5  ^^ 


PRINCIPLES  and  METHODS 


iuujjwuimiw*imm*J'.*ja*i! 


or 


Arithmetic  Teaching 


By  WILLIAM  B.  CHRISWELL,  Ph.  B. 


Published  and  Copyrighted  by  the  Author 

Potsdam,  N.   Y 

1914 

3  V  ,     1 


COPYRIGHT,  1914 
By  William  B.  Chriswell 


Potsdam,  N.  Y. 
Elliot  Fay  &  Sons 

1914 


A 
\  :- 

C 


PREFACE 

The  following  pages  have  been  prepared  for  the 
use  of  students  in  the  Potsdam  State  Normal  School. 

Thanks  are  due  to  Professor  L.  D.  Taggart, 
Superintendent  of  the  Training  School,  for  his  kind- 
ness in  reviewing  the  manuscript  and  offering  valuable 
suggestions. 

VV.  B.  C. 


PART  ONE 


GENERAL  PRINCIPLES. 

It  has  been  well  said  that  we  should  be  loyal  to  principles  but 
the  slave  to  no  man's  devices.  Only  a  few  of  the  general  principles 
especially  applicable  to  Arithmetic  teaching  can  be  treated  in  this 
course,  and  these  but  briefly.  For  fuller  treatment  the  student  is 
referred  to  Younr?,  The  Teaching  of  Mathematics;  Smith,  The  Teach- 
ing of  Elementary  Mathematics;  Suzzallo,  The  Teaching  of  Primary 
Arithmetic;  McMurry,  Special  Method  in  Arithmetic;  McLellan  and 
Dewey,  The  Psychology  of  Number;  Brooks,  Philosophy  of  Arithme- 
tic; De  Garmo,  Interest  and  Education;  and  general  works  on  Psychol- 
ogy and  Education. 

I.   REASONS   FOR  THE  STUDY  OF  METHOD. 

It  is  not  so  much  whether  a  teacher  is  a  good  teacher  as  whether 
she  is  a  progressive  teacher.  The  good  teacher  may  deteriorate,  the 
progressive  teacher  never.  Proper  method  is  as  essential  to  the 
teacher  as  to  the  singer  or  the  painter.  Without  some  knowledge  of 
the  technique  of  the  profession,  little  progress  can  be  made. 

Teachers  may  be  classified  as  follows: 

I.  UNINSPIRED. 

1.  The  unskilled  are  those  who  have  neither  natural  ability  for 
teaching  nor  knowledge  of  technique.  The  sooner  such  are  eliminated 
form  the  profession  the  better. 

2.  The  skilled  are  those  who  have  little  natural  talent  for  teach- 
ing, but  who  have  through  the  study  of  method  mastered  the  tech- 
nique of  the  profession.  These  are  good  teachers  but  lacking  in 
inspiration. 

II.  INSPIRED. 

1.  The  born  teacher  is  full  of  enthusiam  and  inspiration,  but 
lacking  in  knowledge  of  technique.  Her  methods  are  often  faulty, 
but  this  is  more  than  counterbalanced  by  the  inspiration  imparted  to 
others. 

2.  The  artists  are  the  born  teachers  who  have  become  masters 
of  technique.  They  are  the  Michael  Angelos,  the  Rembrandts,  the 
Millets  of  the  teaching  profession. 

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The  purpose  of  the  study  of  method  is  to  transform  into  skilled 
teachers  those  who  would  otherwise  be  unskilled,  and  into  artists 
those  who  are  born  teachers.     To  quote  from  Froebel: 

"Step  by  step  lift  bad  to  good; 
Without  halting,  without  rest, 
Lifting  better  up  to  best." 

See  Young,  Chap.  I,  VIII,  and  pp.  254  and  256;  Page,  Theory  and 
Practice  of  Teaching,  pp.  19-29;  Suzzallo,  Chap.  Ill;  Young,  Teaching 
of  Mathematics  in  Prussia,  Chap.  IV. 

II.  AIMS  OF  ARITHMETIC  TEACHING. 

Various  aims  of  arithmetic  teaching  have  been  given,  the  chief 
being  intellectual  training,  and  utilitarian  purposes.  The  reasoning 
connected  with  the  solution  of  problems  fulfills  the  intellectual  aim. 
For  practical  use  the  essentials  are  knowledge  of  number  facts;  fund- 
amental operations  in  integers,  fractions  and  decimals;  practical 
problems  including  percentage;  and  absolute  accuracy  and  a  resason- 
able  degree  of  rapidity.  Accuracy  is  the  sine  qua  non. .  "Careless 
facility",  says  Professor  Frank  Hall,  "is  not  merely  useless,  it  is  posi- 
tively harmful."  See  Smith,  Chap.  II;  Young,  Chap.  II;  and  pp.  203 
to  209,  214  to  216;  Suzzallo,  Chap  II;  Home,  Philosophy  of  Education, 
pp.  115,  116;  Spencer,  Education,  pp.  29-31;  Brown  and  Coffman,  How 
to  teach  Arithmetic,  Chap.  III. 

III.   FUNCTION  AND  EXTENT  OF  OBJECTIVE  TEACHING. 

Clear  perception  is  essential  as  a  basis  for  future  knowledge.  At 
first  new  ideas  must  be  presented  directly  to  the  senses.  Later  these 
ideas  may  be  used  as  a  foundation  for  an  "apperceptive  system."  To 
produce  clear  knowledge  in  any  science  in  its  initial  stages  requires 
objective  presentation.  The  extremes  of  no  objective  illustration  and 
?ndless  illustration  should  be  avoided.  A  safe  guide  is  the  old  rule  to 
proceed  from  the  known  to  the  related  unknown.  When  there  is  no 
known,  objective  illustration  is  essential.  If  there  is  clear  knowledge 
from  which  to  proceed,  it  is  a  waste  of  time  and  a  loss  of  power  not 
to  use  such  knowledge.  See  Young,  pp.  107,  209  to  212;  Smith,  pp. 
101  to  100;  McLellan  and  Dewey,  p.  283;  Bagley,  Educative  Process, 
PP.  254,  255;  Suzzallo,  pp.  42-59. 

IV.   ESSENTIALS   FOR    EFFICIENT   RECALL. 

To  commit  to  memory  and  to  be  able  to  recall  our  knowledge 
when  needed,  certain  mental  laws  must  be  observed.  Among  the 
essentials  are  strong  impression,  association,  repetition,  and  frequent 
recall. 

Few  facts  in  arithmetic  stand  out  so  prominently  in  and  of  them- 
selves as  to  make  a  strong  and  lasting  impression  at  the  first  present- 

5 


ation.  One  fact  is  as  important  as  another.  Hence  little  use  can  be 
made  of  strong  impressions.  However,  as  the  more  concentrated  the 
attention,  the  stronger  the  impression,  the  teacher  must  require  the 
closest  possible  attention  from  every  child.  As  the  time  of  commit- 
ting number  facts  is  ordinarily  during  the  age  of  passive  or  involun- 
tary attention,  and  little  reliance  can  be  placed  on  forced  attention 
the  teacher  must  appeal  to  the  child's  instincts.  She  must  appeal  to 
his  curiosity,  satisfy  his  desire  for  new  knowledge,  for  variety,  for 
activity,  for  rhythm,  for  success.  By  means  of  games  she  may  associ- 
ate the  work  with  child  life.  >/ 

Drill,  repetition,  and  frequent  recall  will  be  found  necessary.    All 
facts  must  be  fully  learned  even  at  the  cost  of  much  drudgery.     "It  is 
safe  to  say  that  the  point  will  never  be  reached  where  pain  and  drudg- 
ery can  be  entirely  eliminated  from  the  educative  process",  says  Bag- 
ley.     "If  the  pupil  does  not  sometimes  find  his  school  work  disagree- 
able, then  something  is  radically  wrong  either  with  the  pupil  or  with 
the  school  or  with  both."     See  Young,  pp.  91  to  96.  135,  136,  251;  Mun- 
scerberg,  Psychology  and  the  Teacher,  Chap.  XVI,  and  p.  122;  De  Gar- 
mo,   Essentials  of  Method,   p.   60;  James,   Psychology,   Chap.   XVIII 
Radestock,  Habit,  pp.  32,  74;  White,  School  Management,  pp.  160-166 
Bagley,  op.  cit.,  Chap.  XI;  Smith,  Systematic  Methodology,  pp.  37,  38 
Chamberlain,  The  Child,  pp.  341,  342. 

V.  TEACHING  A  NEW  NUMBER  FACT— DRILLS. 

In  teaching  a  new  number  fact,  as  many  different  cortical  areas 
as  possible  should  be  approached  simultaneously  or  in  quick  succes- 
sion, in  order  to  make  the  proper  nerve  fiber  connections  in  the  brain 
as  a  basis  for  future  association  of  ideas  and  quick  recall.  Approach 
visual,  auditory,  vocal  muscular,  and  graphic  muscular  (strain)  areas. 

1.  Teacher  repeat. 

2.  Children  repeat. 

3.  Teacher  write  on  board. 

4.  Children  read  orally. 

5.  Teacher  erase  and  children  visualize. 

6.  Children  repeat  imaged  fact. 

7.  Children  copy  written  fact. 

8.  Children  write  imaged  fact. 

Repeat  these  processes  to  "form  the  habit"  as  it  is  commonly 
expressed,  but  speaking  with  physical  accuracy,  to  form  the  proper 
nerve  fiber  connections.  Have  class  repetition  interspersed  with  in- 
dividual repetition.  Drill  individuals  found  slow,  then  the  class 
again.  Give  very  slow  pupils  outside  drill.  No  other  work  of  any 
kind  should  be  taken  up  after  presenting  new  facts  till  these  have 
been  thorougly  drilled  upon.     As  already  said  all  facts  must  be  fully 


learned  even  at  the  cost  of  much  drudgery.     Give  a  minute's  absolute 
rest,  then  drill  again. 

The  teacher  need  not  be  surprised  if  the  last  facts  presented  are 
forgotten  before  the  next  recitation.  She  must  repeat  the  process 
and  drill  again.  As  often  as  it  is  found  that  some  fact  has  been  for- 
gotten, drill  must  again  be  given  on  this  particular  fact.  Before  pre- 
senting new  facts,  be  sure  that  the  class  know  thoroughly  all  previ- 
ously presented  facts.  Find  what  facts  a  child  does  not  know  and 
drill  him  accordingly.  Responses  should  be  practically  instantaneous. 
"A  child  must  not  be  allowed  to  forget  old  facts,"  says  Munsterberg, 
"in  the  effort  to  acquire  new  facts."  See  Thompson,  Brain  and  Per- 
sonality; Munsterberg,op.  cit.,  pp.  117,  118,  140-147;  Suzzallo,  Chap. 
VIII;  Bailey,  Teaching  Arithmetic,  p.  47;  Bagley,  pp.  122,  123,  328- 
331;  McMurry,  pp.  46,  56;  James,  Psychology,  pp.  134-150,  298; 
Myers,  Experimental  Psychology,  pp.  72-90,  112-116;  Brown  and 
Coffman,  op.  cit.,  Chap.  VIII. 

VI.  REVIEWS. 

We  have  noted  that  one  of  the  essentials  to  memory  is  frequent 
recall.  This  is  one,  but  only  one,  of  the  aims  of  reviews.  There  are 
six  purposes  of  reviews. 

1.  That  the  teacher  may  learn  the  child's  needs. 

2.  To  show  the  child  his  needs. 

3.  To  fix  more  firmly  in  mind  (by  frequent  recall). 

4.  To  organize  materials — the  general  review. 

5.  As  a  preparation  for  new  related  knowledge — the  first  step 

of  the  Inductive  Development  Lesson. 

6.  To  make  knowledge  usable.      This,  says  Gordy,  is  the  great 

function  of  reviews.      (Lessons  in  Psychology.) 

According  to  Gordy  there  are  three  stages  of  knowing: 

1.  Implicit  knowledge — The   child   knows     but   cannot   express 

what  he  knows. 

2.  Explicit  knowledge — The  child  can  tell  what  he  knows,   but 

he  cannot  use  his  knowledge. 

3.  Usable  knowledge. — The  child  can  use  his  knowledge  as  well 

as  tell  what  he  knows. 

To  show  the  importance  of  reviews,  we  may  make  use  of  the 
illustration  of  the  snow  storm. 

On  a  winter  morning  we  look  out  upon  the  streets  to  see  that  a 
great  depth  of  snow  has  fallen.  We  watch  the  first  team  plod  labo- 
riously through  to  open  a  way.  Another  team  follows,  also  with 
great  difficulty,  but  each  succeeding  team  finds  the  task  easier,  till  at 
length  the  sleighs  glide  past  with  the  greatest  ease.  But  another 
storm  fills  up  the  broken  way.     The  process  of  breaking  out  must  be 

7 


repeated,  but  the  difficulty  is  not  so  great  as  it  was  the  first  time  and 
soon  the  sleighing  is  better  than  before.  So  storm  after  storm  re- 
quires constant  travel  to  keep  the  way  open.  If  after  the  way  has 
been  once  broken  out,  several  storms  occur  without  the  street's  being 
used  in  the  meantime,  the  path  may  become  utterly  obliterated  and 
forgotten. 

So  it  is  with  the  brain.  Children,  like  adults,  will  forget  when 
facts  are  allowed  to  remain  unused  for  some  time.  The  way  must  be 
broken  out  and  rebroken;  or  to  speak  literally,  the  nerve  fiber  connec- 
tions must  be  strengthened  by  repeated  use. 

Drills,  therefore,  should  be  given  not  only  in  connection  with 
new  work,  but  also  in  reviews.  Drills  should  be  quick  and  snappy. 
In  review  drills  when  the  child  hesitates,  call  on  the  class.  Work 
rapidly  while  you  work  and  waste  no  time.  Ten  minutes  of  rapid  fire 
recitation  is  worth  more  than  a  half  hour  of  dawdling. 

See  Munsterberg,  op.  cit.,  p.  122  and  Chap.  XVI;  Smith,  pp.  143, 
144;  Young,  pp.  129,  142,  225,  226;  McMurry,  pp.  56,  80,  81,  98,  127, 
136,  137,  140;  McMurry,  Method  of  the  Recitation,  pp.  114,  115; 
Radestock,  Habit,  p.  21;  Bagley,  op.  cit.  pp.  328-333. 

VII.  CONCERT  RECITATIONS. 

Beware  of  concert  recitations.  A  skillful  teacher  may  at  times 
use  concert  recitations  successfully. 

1.  In  drill  work.  Here  what  is  desired  is  repetition  with  atten- 
tion, hence  time  is  saved  by  having  several  repeat  simultaneously. 

2.  When  the  answer  is  short  and  one  child  fails,  the  rest  of  the 
class  may  have  the  answer  at  their  tongue's  end. 

3.  When  the  answer  is  short  and  a  pause  is  made  after  the 
question  to  allow  all  to  grasp  the  meaning  and  form  the  correct  an- 
swer. In  recitation  on  number  facts,  for  example,  the  teacher  may 
point  to  successive  facts  on  the  board,  pause  briefly,  say  "class"  or 
tap  on  the  board  with  the  pointer,  and  the  class  as  a  result  of  the 
pause  is  ready  to  answer  simultaneously.  If,  however,  no  time  is 
given  for  thought,  one  bright  child  will  answer  and  the  rest  then  re- 
peat like  parrots,  some  not  even  looking  at  the  board  to  see  what  the 
fact  is. 

Where  there  is  a  possibility  of  different  wordings  to  the  answer, 
concert  recitation  is  out  of  the  question.  See  Page,  Theory  and 
Practice  of  Teaching,  pp.  151,  152. 

VIII.   ATTENTION. 

Keep  every  child  attentive  as  long  as  he  is  supposed  to  be  atten- 
tive. If  impossible  to  hold  the  attention  of  the  more  advanced  pupils 
while  drilling  backward  pupils,  set  them  at  other  work.  Do  not  try 
to  keep  young  children  at  high  tension  too  long  at  a  time.  Allow 
them  to  rest,  then  begin  again. 

8 


TO  HOLD  ATTENTION. 

1.  Secure  interest  if  possible.  In  some  way  connect  the  work 
with  child  life,  (lames  are  often  a  great  aid.  But  be  sure  to  hold 
attention,  interest  or  no  interest.  One  of  the  great  aims  of  education 
is  the  development  of  the  power  of  voluntary,  or  active,  attention. 
"May  it  not  be",  asks  Munsterberg,  "that  the  most  important  aim  of 
education  is  just  the  power  of  overcoming  the  temptations  of  mere 
personal  interest?"  "An  education"',  he  says,  "which  simply  follows 
the  likings  and  interests,  leaves  the  adolescent  personality  in  a  flabby 
and  ineffective  state."  "There  is  no  doubt  that  through  the  tendency 
of  our  times  to  yield  to  this  demand  for  interesting  instruction,  we 
already  feel  the  dangerous  results  of  the  crippling  of  the  voluntary 
attention."  "Public  life  has  to  suffer  for  it."  "But  the  great  word 
which  is  to  control  it  is  not  pleasure  but  duty."  Op.  cit.,  pp.  16,  18, 
190,  265,  269. 

On  the  other  hand,     George  Eliot  says:  For  getting  a  fine 

flourishing  growth  of  stupidity  there  is  nothing  like  pouring  out  on  a 
mind  a  good  amount  of  subjects  in  which  it  feels  no  interest.  Mill  on 
the  Floss,  p.  327. 

2.  Watch  the  class,  not  the  pupil  reciting. 

3.  Call  on  inattentive  pupil  when  pupil  reciting  hesitates. 

4.  Call  on  the  whole  class  when  the  pupil  reciting  hesitates, 
provided  the  answer  is  brief.  If  but  few  respond,  it  may  be  a  sign 
of  inattention.  See  De  Garmo,  Chap.  XV;  Munsterberg,  op.  cit.,  Chap. 
XVIII;  James,  Psychology,  Chap.  XIII;  Radestock,  Habit,  pp.  70,  71: 
Puffer,  Psychology  of  Beauty,  pp.  160-168;  Bagley,  Classroom  Man- 
agement, pp.  137-187. 

IX.  THE  ART  OF  QUESTIONING. 

No  teacher  of  Arithmetic  should  fail  to  make  a  thorough  study 
of  the  art  of  questioning.  The  mere  instructor  has  little  need  of 
questioning,  but  the  true  teacher  must  know  how  to  question  and 
how  to  question  well.  "To  question  well  is  to  teach  well,"  says  De- 
Garmo.  Those  desiring  to  make  themselves  more  proficient  in  this 
art  are  referred  to  the  chapter  on  "The  Art  of  Questioning"  in  De- 
Garmo's  "Interest  and  Education".     Also  see  Young,  pp.  55,  67. 

X.   CLASS  V.  S.  INDIVIDUAL  RECITATION. 

The  teacher  of  power,  like  the  conductor  of  the  trained  choir, 
carries  along  his  whole  class.  Here  may  be  a  solo  and  there  a  duet 
followed  by  the  ensemble,  all  being  parts  of  one  great  production. 
So  the  teacher  may  call  on  this  individual  or  on  that  to  recite  for  the 
class  and  as  a  constituent  part  of  the  class,  but  the  whole  class  is 
carried  along  as  if  it  were  a  single  individual.  Every  one  is  attentive 
and  ready  to  respond  if  called  upon.  The  individual  is  by  no  means 
neglected.     The  teacher  watches  every  move,  every  expression  even. 


The  slightest  sign  of  incomprehension  in  a  child's  eye  calls  forth  an 
enlightening  illustration  or  a  thoughtprovoking  question. 

However  this  complete  class  unity  is  found  neither  in  the  first 
recitation  nor  in  the  tenth.  More  or  less  individual  drill  must  pre- 
cede and  accompany  the  choir  practice.  The  quick  ear  of  the  conduc- 
tor catches  every  weak  point,  and  if  he  cannot  strengthen  it  in  gen- 
eral practice,  he  does  it  in  private.  So  the  successful  class  room 
teacher  must  supplement  his  class  room  work  with  individual  instruc- 
tion. This,  in  fact,  is  the  strength  of  the  Batavia  System,  in  which 
one  teacher  does  the  class  instruction  and  another  the  individual 
instruction. 

But  one  might  visit  two  adjoining  rooms  and  hear  the  same 
question  put  and  practically  the  same  answer  given  by  individual 
pupils;  yet  in  one  room  the  recitation  might  be  class  recitation  and  in 
the  other  individual  recitation.  That  is,  in  the  one  room  the  recita- 
tion might  be  conducted  in  the  manner  described,  while  in  the  other 
only  the  person  reciting  may  be  paying  attention.  The  class  as  a 
whole  is  receiving  no  benefit  and  the  recitation  consists  of  a  series  of 
individual  recitations,  logically  connected  in  the  teacher's  mind  per 
haps,  but  disconnected  and  unrelated  in  the  pupils'  minds.  See 
Young,  56,  57,  81  to  86,  138,  139;  Bagley,  Classroom  Management 
Chap.  XIV;  Young,  The  Teaching  of  Mathematics  in  Prussia,  pp.  56, 
57. 

XI.  THE  INDUCTIVE  DEVELOPMENT  LESSON. 

Inductive  reasoning  consists  in  passing  from  particular  truths 
to  general  truths.  It  is  known  as  generalization.  Induction  is  often 
the  method  used  in  presenting  new  matter  to  children.  The  child,  in 
fact,  begins  his  acquisition  of  knowledge  by  means  of  induction. 

The  inductive  development  lesson  is  generally  considered  as 
having  five  steps: 

1.  Preparation. 

Substep — Statement  of  the  aim. 

2.  Presentation. 

3.  Comparison  and  Abstraction. 

4.  Generalization. 

5.  Application. 

The  preparation  consists  of  a  brief  review  of  all  matter  necessary 
as  a  basis  for  the  new  work  to  be  presented,  in  order  that  the  child 
may  have  this  knowledge  fresh  in  his  mind  and  thus  readily  take  the 
step  from  the  known  to  the  related  unknown.  Not  the  whole  subject 
should  be  reviewed,  but  only  that  necessary  for  the  foundation. 
Generally  a  few  questions  will  bring  out  all  that  is  required.  See 
McMurry,  pp.  76,  79,  80;  DeGarmo,  Essentials  of  Method,  pp.  46-51. 

As  to  the  statement  of  the  aim,  De  Garmo  says:  "It  would  be 
unpedagogical  not  to  have  the  pupil  understand  from  the  beginning 
what  the  aim  of  the  lesson  is."      "A  skillful  dramatist  never  fully 

10 


1 


reveals  his  plot  ahead  of  its  unfolding,  nor  does  he,  on  the  other  hand 
ever  allow  any  great  but  entirely  unexpected  culmination  to  occur." 
Essentials  of  Method,  pp.  48,  50.  "It  should  state  as  clearly  as  pos- 
sible the  point  that  the  lesson  is  intended  to  make,"  says  Bagley.  "It 
should  seize  upon  some  need  and  show  it  may  be  satisfied."  "The 
aim  really  forms  the  connecting  link  between  the  old  and  the  new." 
Op.  cit.,  pp.  291,  292.  See  McMurry  80,  81.  However,  many  teachers 
of  successful  experience  deliberately  omit  this  sub-step. 

The  presentation  consists  in  placing  before  the  class  several 
illustrations  of  the  new  matter  to  be  taught,  reaching  it  naturally 
from  the  matter  reviewed  in  the  preparation.  The  new  matter  may 
be  brought  out  by  questions,  but  occasionally  it  may  be  found  best  to 
show  the  connection  directly  without  questions.  See  De  Garmo,  op. 
cit.,  pp.  51,  52. 

The  comparison  and  abstraction  consists  in  noting  what  has  been 
done  in  all  examples. 

The  generalization  consists  in  stating  in  the  form  of  a  general 
rule  or  formula  how  all  similar  examples  should  be  worked.  See 
McMurry,  p.  69. 

The  application  consists  in  working  examples  by  the  rule  that 
has  been  generalized.  This  step  as  will  be  seen  is  deductive,  though 
a  part  of  the  inductive  development  lesson.  See  Smith,  pp.  Ill,  112; 
Bagley,  op.  cit.,  Chap.  XIX.  For  lesson  plans,  see  Stamper,  A  text 
book  on  the  teaching  of  Arithmetic,  Chap.  VIII. 

XII.  TEACHING  A  NEW  OPERATION. 

In  teaching  a  new  operation,  when  possible,  use  the  inductive 
process  as  above.  But  aside  from  this,  when  the  class  is  large,  there 
is  a  general  method  of  procedure  that  will  often  be  found  advanta- 
geous. 

1.  In  developing  the  process  the  teacher  should  work  several 
easy  examples  on  the  board. 

2.  Send  individuals  to  the  board,  the  class  observing  and  mak- 
ing suggestions  under  the  direction  of  the  teacher. 

3.  Send  several  pupils  or  the  whole  class  to  the  board. 

4.  If  several  have  trouble,  send  all  to  their  seats  and  attack  the 
difficulty  again. 

5.  Send  the  class  to  the  board  again  to  finish  the  example. 

Or  in  place  of  4  and  5,  as  for  example  in  teaching  long  division, 
the  teacher  may  ask  the  class  to  work  as  she  directs.  Suppose  each 
child  has  on  the  board  ready  for  work  the  example  384  to  be  divided 
by  12.  Teacher — How  many  times  is  12  contained  in  38?  Where 
shall  we  place  the  3?  What  shall  we  do  with  the  3  now?  Where 
shall  we  place  the  36?  Etc.  Several  examples  may  be  worked  in  this 
way  until  the  class  know  all  the  steps. 

6.  Teacher  give  individual  help  to  the  few  who  still  have  trouble, 

11 


7.  Slow  pupils  should  be  dealt  with  out  of  class  to  prevent  the 
monopolizing  of  time  that  should  be  spent  in  class  instruction. 

It  is  important  that  but  one  difficulty,  that  is  one  new  point, 
should  be  introduced  at  a  time.  "The  importance  of  cutting  work  up 
into  simple  steps  and  taking  them  one  at  a  time",  says  Young,  "can- 
not be  overestimated."     (p.  128.)     See  Young,  pp.  134,  135. 


XIII.  THE   DEDUCTIVE   DEVELOPMENT  LESSON. 

Deductive  reasoning  consists  in  passing  from  a  general  truth  to 
a  particular  truth.  We  begin  with  rules  and  proceed  to  individual 
cases.  Our  first  duty  is  to  see  that  the  major  premise  is  true.  In 
arithmetic  we  must  apply  the  proper  rule. 

In  Logic  we  have  the  Major  Premise,  the  Minor  Premise,  and 
the  Conclusion. 

Major  Premise:  All  A  is  B.  Minor  Premise:  C  is  A.  Conclu- 
sion: C  is  B. 

The  Deductive  Development  Lesson  is  considered  as  made  up  of 
four  steps: 

1.  Statement  of  data.  2.  Statement  of  governing  principles  or 
general  rule.     3.  Statement  of  the  conclusion.     4.  Verification. 

It  will  be  noted  that  the  order  of  the  logical  sylogism  and  of  the 
deductive  development  lesson  as  sometimes  given  is  here  reversed, 
the  particular  statement  being  given  before  the  general.  In  promis- 
cuous problem  solving  in  arithmetic  this  inverted  order  is  necessary, 
as  the  data  is  first  stated  and  the  pupil  must  determine  what  general 
rule  applies  to  the  particular  case.  He  then  draws  his  conclusion, 
states  the  problem  accordingly,  and  solves.  The  Verification  may  b-" 
some  check  or  reverse  process,  but  it  frequently  consists  in  looking  at 
the  answer  or  in  receiving  the  instructor's  approval. 

Example:  A  merchant  invests  $160  in  flour  and  sells  it  at  a 
profit  of  $40.     Find  the  Rate  of  gain. 

1.  Statement  of  data:  $160  is  Cost  or  B.  $40  is  gain  or  P. 
Wanted — the  Rate  of  gain  or  R. 

2.  Statement  of  general  rule:     P  divided  by  B  =  R. 

3.  Statement  of  conclusion:  $40  h- $160  =  25 'v .  Rate  of  gain. 
Answer.  % 

4.  Verification:     25  #    of  $160  =  $40. 

See  Bagley,  Educative  Process,  Chap.  XX. 

XIV.   PROBLEM  SOLVING. 

There  are  several  difficulties  that  the  child  will  have  to  meet  and 
remove  before  he  can  solve  problems  readily. 

1.  He  must  be  able  to  identify  the  data  and  what  is  wanted. 

2.  He  must  determine  what  rule  or  formula  to  use. 


3.  He  must  determine  what  intermediate  steps,  if  any,  must  1"' 
taken.     Under  this  head  he  must  determine: 

(1)  Is  there  a  missing  fact  and  what  is  it? 

(2)  Can  the  missing  fact  be  found  directly  from  the  given 
data? 

(3)  If  the  fact  cannot  be  thus  found,  what  can  be  found 
from  the  given  data,  from  which  new  data  the  missing 
fact  may  be  found?  This  is  an  appeal  to  the  puzzle 
instinct. 

Much  practice  in  identification  of  data  in  terms  of  Percentage. 
Mensuration,  etc.,  should  be  given,  independent  of  solution.  At  first 
it  will  be  necessary  to  lead  to  identification  by  questioning.  Then 
the  child  should  be  led  to  ask  himself  the  questions.  For  a  time  he 
should  be  required  to  write  on  paper  or  on  the  board  the  data  both  in 
terms  of  the  problem  and  in  terms  of  Percentage,  if  that  is  the  topic 
under  study. 

Even  greater  than  identification  of  data  may  be  the  difficulty  of 
determining  and  finding  the  missing  fact. 

Problem:  If  4  hats  cost  $12,  what  will  7  hats  cost?  Here  the 
missing  fact  is  the  cost  of  one  hat.  Question  as  to  what  must  first 
be  known  before  we  can  find  the  cost  of  7  hats.  Can  we  find  that 
missing  fact?     How' 

The  following  problem  of  exactly  the  same  nature  as  the  above 
sometimes  puzzles  eighth  grade  pupils:  If  %  of  a  bushel  of  wheat 
cost  *'\  of  a  dollar,  what  will  -'  :!  of  a  bushel  cost?  Using  the  former 
problem  as  a  preparation  determine  the  rule:  divide  the  cost  by  the 
number  to  find  the  cost  of  one,  and  multiply  this  by  the  number  of 
which  the  cost  is  desired.     In  both  problems  solve  by  cancellation. 

Problem:  What  will  17  gallons  of  milk  cost  at  6  cents  per  quart? 
Here  the  essential  missing  fact  may  be  given  as  either  the  price  per 
gallon  or  the  number  of  quarts.  For  further  illustration,  see  Chapter 
on  Percentage. 

XV.  TEACHING  HOW  TO  STUDY. 

One  of  the  greatest,  if  not  the  greatest  end  of  intellectual  edu- 
cation, is  to  teach  the  child  how  to  study  independently  of  a  teacher. 
Sometimes  it  seems  that  altogether  too  much  teaching  is  being  done 
and  the  pupil  is  becoming  entirely  dependent  upon,  instead  of  inde- 
pendent of,  his  teacher. 

"Dependence  is  not  the  preparation  for  independence-',  says 
'lurry.  "Indeed,  preat  skill  on  the  part  of  a  teacher  in  these 
respects  almost  precludes  such  skill  on  the  part  of  pupils.  If  allow- 
ed prominence  year  after  year,  it  so  undermines  self-reliance  that 
one's  helplessness  when  alone  is  greatly  increased."  "By  overlook- 
ing the  difference  between  studying  with  a  leader  and  alone,  there- 
fore,  the  teacher   overlooks   initiative,    and   in    consequence,    she   not 

13 


only  fails  to  develop  that  power,  but  she  may  easily  undermine  it  by 
accustoming  her  pupils  to  dependence  upon  her."  (How  to  study  and 
teaching  how  to  study,  p.  290.) 

Too  much  has  the  appeal  been  made  to  the  ear.  The  child  can 
make  little  or  nothing  from  his  text  book  without  the  teacher's  first 
clearing  up  all  difficulties  for  him  in  the  "assignment".  This  is  a 
fine  thing  for  the  dull  pupil,  but  the  bright  child  is  weakened.  Many 
of  us  were  taught  by  the  other  extreme.  The  text  book  was  placed 
in  our  hands  and  we  were  compelled  to  dig  out  the  assigned  lesson  as 
best  we  could.  This  was  fine  for  the  strong,  but  woe  be  unto  the 
weak.  If  he  could  not  work  an  example  and  was  compelled  to  call  on 
his  teacher,  the  latter  would  merely  work  out  the  example  on  a  slate 
and  return  it  without  a  word  of  explanation.  A  medial  course  is  the 
best  to  follow.  At  first  give  sufficient  assistance  by  means  of  sug- 
gestion and  illustration,  gradually  decreasing  the  amount  till  the 
child  is  largely  thrown  upon  his  own  responsibility. 

A  place  where  the  child  should  early  gain  independence  is  in  the 
solution  of  problems.  This  should  first  be  taught  by  means  of  the 
Study  Recitation,  given  for  a  time  once  a  week  perhaps.  On  other 
days  the  difficulties  may  be  cleared  up  in  the  assignment  by  means 
of  questions  and  illustrations  as  already  indicated.  But  the  child 
should  be  taught  as  soon  as  possible  to  ask  himself  the  necessary 
questions  without  the  aid  of  the  teacher.  "Power  of  initiative  is  the 
key  to  proper  study",  says  McMurry.  Op.  cit.,  p.  288.  See  Page,  op. 
cit.,  p.  45. 

XVI.  THE   STUDY  RECITATION. 

The  study  recitation  is  at  first  given  after  the  method  of  the 
deductive  development  lesson.  The  problems  should  for  a  time  be 
placed  on  the  board,  but  at  length  the  regular  text  book  lesson  should 
be  taken. 

1.  The  children  read  the  problem  silently. 

2.  Children  write  on  paper  a  statement  of  data  and  what  is  to 
be  found.  If  in  Percentage,  identify  in  terms  of  Percentage.  In  the 
early  lessons  the  teacher  will  find  it  necessary  to  question  as  to  what 
each  item  in  the  problem  is,  for  a  time  even  having  the  answer  given 
orally.  Then  the  questions  will  be  merely  asked  by  the  teacher  and 
answered  mentally  unless  some  pupil  indicates  that  he  cannot  identi- 
fy a  particular  item.  The  teacher  may  ask  how  many  can  identify 
this  item  or  that,  or  who  cannot  identify  it;  or  she  may  merely  ask 
whether  there  is  anyone  who  cannot  identify  all  the  items.  The 
teacher's  questioning  should  become  less  and  less,  the  children  asking 
themselves  the  questions  and  gradually  becoming  more  and  more  self 
reliant. 

The  teacher  finally  does  nothing  but  pass  among  the  pupils, 
glancing  at  their  work  and  putting  a  question  here  and  there  as  she 
finds  the  necessity.     In  time  the  pupils  will  take  this  step  mentally 

14 


^/ 


and  put  nothing  on  their  papers,  the  great  danger  being  that  they 
will  do  so  before  they  are  strong  enough.  But  this  must  be  the  ulti- 
mate aim. 

3.  Children  determine  how  to  find  the  result,  whether  it  can  be 
found  directly  or  whether  some  intermediate  step  is  required.  This 
is  the  step  in  which  the  reasoning  must  take  place.  As  in  the  former 
step  the  teacher  will  have  to  direct  thought  by  means  of  questions. 
How  do  you  find  the  result?  Can  you  find  it  directly?  What  is  the 
essential  missing  fact?  Can  you  find  that?  What  must  you  find 
first?  Etc.,  etc.  As  before,  the  child  should  gradually  learn  to  do  his 
own  questioning,  the  teacher's  suggestions  becoming  rare. 

4.  The  children  state  the  problem. 

5.  Solve  and  place  result  in  the  statement. 

6.  Verify  by  some  check  if  possible. 

The  Study  Recitation  should  be  given  during  tfte  regular  recita- 
tion period.  For  a  time  twice  or  at  least  once  a  week  should  be  given 
to  such  work.  Though  it  may  seem  impossible  to  take  so  much  time 
from  the  ordinary  recitation,  yet  in  the  end  it  will  be  found  that  much 
time  has  been  saved. 

When  the  child  is  able  undirected  to  read  and  solve  miscella- 
neous problems  in  Arithmetic,  he  has  made  a  long  step  in  the  direc- 
tion of  one  of  the  chief  aims  of  school  education-,  ability  to  educate 
himself  from  books  without  the  actual  bodily  presence  of  a  teacher. 
He  is  becoming  a  student.  "When  the  principles  can  be  explicitly 
stated  and  intelligently  applied,  the  essential  aim  of  arithmetic  has 
been  reached."  "The  intellectual  treasures  of  the  past  lie  locked  up 
in  books..  Proper  school  training  unlocks  this  storehouse  by  accus- 
toming one  to  their  intelligent  use."  (McMurry,  The  Method  of  the 
Recitation,  pp.  6,  142,  143.) 

.  See  Bagley,   op.  cit.,  pp.   316-322;   Bagley,   Classroom   Manage- 
ment, pp.  206-210. 

XVII.  THE  ASSIGNMENT. 

By  the  Assignment  as  used  by  writers  on  pedagogy  is  meant  not 
merely  the  lesson  assigned  for  home  study  but  the  method  of  assign- 
ing the  lesson.  Bagley  says:  "This  is  a  preliminary  clearing  of  the 
road  before  the  seat  work  begins."  "The  acme  of  a  skillful  assign- 
ment is  reached  when  the  teacher  reveals  just  enough  of  what  is  con- 
tained in  the  lesson  to  stimulate  in  the  pupils  the  desire  to  ascertain 
the  rest  for  themselves."  "In  general,  the  assignment  will  be  much 
more  explicit  and  detailed  in  the  intermediate  grades,  where  the  pupil 
is  just  learning  to  use  text  books,  than  in  the  upper  grades  and  the 
high  school,  where  some  familiarity  with  the  text  book  method  may 
be  assumed.  But  in  all  cases  the  assignment,  whether  it  be  brief  or 
full,  is  an  important  step  which  should  never  be  omitted."  (Op.  cit., 
pp.  317,  318.)  The  purpose  of  the  assignment,  therefore,  is  to  clear 
up  all  insurmountable  difficulties. 

15 


For  example,  suppose  the  lesson  to  be  assigned  consists  of  prob- 
lems and  that  the  class  has  had  little  work  of  the  kind.  The  assign- 
ment should  consist  of  such  questions  as  are  used  in  the  Study  Reci- 
tation. The  study  recitation  prepares  for  this.  Only  a  sufficient 
number  of  questions  should  be  asked  to  enable  the  pupils  to  solve  the 
problems  without  aid  during  the  study  period.  The  ability  of  the 
pupils  to  solve  the  problems  will  show  the  teacher  how  successful  was 
her  assignment.  Some  pupils  will  understand,  or  think  they  under- 
stand, under  the  questioning  of  the  teacher;  but  when  they  come  to 
take  up  their  lesson  alone,  they  find  they  are  unable  to  solve  the 
problem.  Such  pupils  will  need  to  be  requestioned.  As  in  the  study 
lesson,  the  number  of  questions  will  decrease  as  the  pupils  gain  in 
strength.  This  work  should  go  hand  in  hand  with  the  work  of  the 
study  recitation,  and  in  fact  it  is  an  essential  feature  of  the  method 
of  teaching  children  how  to  study. 

One  danger  that  the  inexperienced  teacher  is  apt  to  fall  into  is 
that  in  the  questioning  in  the  study  recitation  and  in  the  assignment, 
she  will  call  on  the  brighter  pupils,  those  who  have  little  need  of  such 
help.  When  the  questions  are  answered  orally,  those  called  upon 
should  be  the  pupils  meeting  with  difficulty.  This  is  very  different 
from  the  method  followed  in  the  regular  recitation  lesson  when  the 
aim  is  to  learn  how  well  the  lesson  has  been  prepared.  In  such  reci- 
tations, good,  bad,  and  indifferent  must  be  called  upon  indiscriminate- 
ly. See  Young,  pp.  132,  133,  147;  De  Garmo,  Essentials  of  Method, 
p.  47;  Bagley,  op.  cit.,  pp.  317-  319;  Bagley,  Classroom  Management, 
pp    192-206. 

XVIII.   DICTATION. 

Have  all  members  of  the  class  take  down   dictation   promptly,    / 
accurately,   and  neatly.      Dictate  slowly  enough   for  all  to  keep   to- 
gether, constantly  increasing  the  speed  till  the  whole  class  can  take 
dictation  rapidly. 

XIX.  ORAL  AND  SILENT  MENTAL  ARITHMETIC. 

A  large  part  of  the  time  given  to  arithmetic  should  be  devoted  / 
to  oral  work  and  to  silent  mental  work  in  which  only  the  answer  is  * 
written.  New  work  should  generally  be  presented  orally  and  the 
first  written  problems  in  each  new  topic  should  be  as  simple  as  the 
oral  problems.  In  written  work  as  few  figures  as  possible  should  be 
used,  short  processes  and  silent  mental  computation  supplementing 
the  written.  As  a  child  works  at  the  board,  have  him  "chalk  and 
talk";  that  is,  state  orally  each  step  as  he  writes  it.  / 

See  Smith,  p.  118;  Young,  pp.  134,  135,  230;  Suzzallo,  pp.  75-78; 
McMurry,  pp.  57,  59,  80,  82,  98,  107-109,  123,  124,  127,  128,  133; 
Bailey,  A  Handy  Book  on  Teaching  Arithmetic,  p.  50*  Walsh,  Meth- 
ods in  Arithmetic,  pp.  25,  27;  Young,  Teaching  of  Mathematics  in 
Prussia,  p.  57-64;  Munsterberg,  op.  cit.,  283-285;  Smith,  The  teaching 
of  Arithmetic,  Chap.  VII.     See  Stamper,  op.  cit.,  pp.  228-233. 


PART   TWO 

Primary   Arithmetic 


I.     FIRST  LESSONS  IN  NUMBER. 

The  first  lessons  in  number  should  consist  in  finding  the  child's 
knowledge  of  indefinite  relative  magnitude  and  exact  number,  and  his 
ability  to  count.  He  should  be  drilled  in  the  use  and  meaning  of  such 
relative  terms  as  short,  shorter,  shortest;  long,  tall;  large,  few,  etc. 

To  find  the  exact  knowledge  of  number,  objects  may  be  held  up 
and  the  child  called  upon  to  tell  how  many  there  are,  or  he  may  be 
asked  to  bring  a  certain  number  of  objects.  He  does  not  know  the 
number  six  till  he  can  both  select  six  objects  from  a  larger  number 
of  objects  and  also  state  correctly  how  many  there  are  when  six 
objects  are  shown  to  him.  The  knowledge  that  the  individual  child 
may  have  upon  entering  the  first  grade  will  depend  upon  his  age,  his 
home  surroundings  and  training,  and  his  kindergarten  training.  First 
grade  teachers  report  varying  degrees  of  knowledge  from  three  to 
more  than  ten.  la  order  to  have  a  starting  point,  the  common  knowl- 
edge of  the  class  must  be  learned,  then  the  teaching  may  begin  at  this 
point. 

There  is  some  difference  of  opinion  as  to  whether  we  should 
teach  the  numbers  as  far  as  ten,  before  teaching  the  figures  repre- 
senting them;  or  whether  we  should  teach  both  number  and  figure  at 
the  same  time.  The  problem  is  largely  determined  for  us,  however, 
with  respect  to  the  first  numbers  at  least,  by  the  child's  learning  the 
numbers  without  knowing  the  corresponding  figures.  The  more 
common  practice  with  regard  to  the  remaining  numbers  is  to  follow 
the  principle  laid  down  by  Quintilian  regarding  the  teaching  of  the 
Roman  alphabet,  that  is,  to  teach  both  name  and  figure  at  the  same 
time. 

The  question  as  to  the  desirability  of  abstract  counting  is  also 
largely  predetermined.  In  school  practice  some  of  our  best  authori- 
ties hold  that  counting,  to  be  of  value,  should  be  concrete  and  should 
follow  the  study  of  definite  number  so  that  when  a  child  can  count  to 
twenty-five,  he  also  knows  twenty-five  objects.  It  does  not  seem  to 
be  a  matter  of  great  moment;  and  whichever  practice  a  teacher  may 
follow,  she  has  good  authority  to  support  her. 

Taking  for  granted  that  we  are  to  follow  the  practice  of  teach- 
ing all  new  numbers  and  the  figures  representing  them  at  the  same 
time,  we  shall  find  that  our  first  lesson  in  exact  number  must  be 
teaching  the  figures  representing  the  numbers  already  known.  This 
may  be  done  by  adapting  the  word  method  of  teaching  reading.  It 
will  doubtless  be  a  saving  of  time  to  co-ordinate  by  teaching  the  writ- 

17 


ten  word  and  the  figure  at  the  same  time.  In  teaching  the  figure  3, 
supposing  the  class  to  be  fully  acquainted  with  the  number  three, 
there  might  be  some  question  as  to  the  necessity  of  presenting  three 
objects.  The  question  is  whether  the  word  "three"  brings  up  a  suffi- 
ciently vivid  concept  of  the  number  three.  The  same  question  is  pre- 
sented in  the  teaching  of  the  word  "cat".  Is  it  essential  to  present  a 
picture  of  a  cat,  or  does  the  word  bring  up  a  sufficiently  vivid  mental 
picture  to  do  away  with  the  need  of  the  physical  picture?  Opinions 
of  course  will  differ,  but  the  question  is  worth  considering.  The  gen- 
eral practice  of  those  who  believe  in  objective  teaching  would  indicate 
a  belief  in  the  necessity  of  the  physical  picture. 

In  teaching  a  figure  for  a  known  number  we  may  hold  up  the 
number  of  objects,  elicit  the  name  (or  ask  the  children  to  bring  three 
objects),  write  the  figure  on  the  board,  have  children  name  the  figure 
and  write  it  and  bring  the  number  of  objects  indicated.     Drill. 

In  teaching  a  new  number,  as  eight,  supposing  that  some  mem- 
bers of  the  class  do  not  know  the  numbers  beyond  seven,  teacher  hold 
up  eight  objects  and  ask  how  many  there  are?  Bring  out  from  those 
who  do  not  know  the  name  that  there  is  one  more  than  seven.  Havn 
some  child  who  knows  tell  how  many  this  is.  Write  the  figure  8  or 
the  board  and  drill  as  before,  giving  special  attention  to  those  wht 
did  not  know  the  number,  and  having  them  bring  8  objects,  etc. 

In  teaching  zero,  teacher  hold  up  her  empty  hand  and  ask  how 
many  objects  she  has  in  it.  The  class  respond,  "not  any".  Teacher 
write  "0"  on  the  board  and  tell  class  that  this  is  how  we  express  "not 
any"  and  that  we  call  it  naught  or  zero.  Drill.  Warning:  Do  not 
say  nor  allow  children  to  say  "ought". 

See  McLellan  and  Dewey,  pp.  144-195;  Brown  and  Coffman,  131- 
150;  Smith,  pp.  112-117;  Walsh,  pp.  33-45. 

II.   NUMBERS  ABOVE  NINE. 

In  teaching  "ten"  use  the  fingers  on  the  two  hands.  Also  use 
splints,  tying  ten  in  a  bundle.  Have  several  children  stand  and  c-.e 
child  tell  how  many  "ten  fingers"  there  are.  Teacher  write  10  on  the 
board  and  have  children  write  it. 

Teacher  holding  up  one  bundle  of  ten  and  one  single  splint,  How 
many  splints  have  I?  As  in  eleven  we  have  one  ten  and  one  single 
one,  we  write  it  thus,  the  left  hand  "one"  tells  us  we  have  one  "ten", 
and  the  right  hand  "one"  tells  us  we  have  a  single  "one",  or  eleven  in 
all.  Similarly  teach  12  and  13.  If  one  ten  and  one  is  written  thus, 
one  ten  and  two  thus,  and  one  ten  and  three  thus;  who  can  tell  how 
we  shall  write  one  ten  and  four?     One  ten  and  five?     Etc. 

Tell  class  that  -teen  means  ten.  If  four  and  ten  is  called  four- 
teen; six  and  ten,  sixteen;  seven  and  ten,  seventeen;  what  shall  we 
call  eight  and  ten?     Nine  and  ten? 

Teacher  holding  up  two  bundles  of  ten,  how  many  tens  have  I? 

18 


Obtain  or  give  name.  Etc.  Similarly  with  twenty-one,  twenty-two, 
and  twenty-three.  If  twenty  and  one  is  called  twenty-one;  twenty 
and  two,  twenty-two;  twenty  and  three,  twenty-three;  what  shall  we 
call  twenty  and  four?  Twenty  and  five?  Etc.  If  twenty-one  is  writ- 
ten thus;  twenty-two,  thus;  and  twenty-three,  thus;  how  shall  we 
write  twenty-four?  Twenty-five?  Etc.  If  one  ten  is  written  thus-, 
two  tens,  thus;  and  three  tens,  thus;  how  shall  we  write  four  tens? 
Etc.  If  four  tens  is  called  forty;  five  tens,  fifty;  and  six  tens,  sixty; 
what  shall  we  call  seven  tens?     Etc. 

If  the  method  of  induction  thus  briefly  sketched  is  used,  the  class 
should  soon  learn  to  name  and  write  new  numbers  without  the  teach- 
er's aid  except  by  means  of  questioning.  As  the  numbers  are  given 
by  the  child,  the  teacher  should  build  up  the  following  table.  By 
having  the  known  numbers  thus  systematically  before  the  child,  he 
will  more  readily  determine  how  to  write  the  new  numbers. 


1 

11 

21 

31 

41 

51 

61 

71 

81 

91 

2 

12 

22 

32 

42 

52 

62 

72 

82 

92 

3 

13 

23 

33 

43 

53 

63 

73 

83 

93 

4 

14 

24 

34 

44 

54 

64 

74 

84 

94 

5 

15 

25 

35 

45 

55 

65 

75 

85 

95 

6 

16 

26 

36 

46 

56 

66 

76 

86 

96 

7 

17 

27 

37 

47 

57 

67 

77 

87 

97 

8 

18 

28 

38 

48 

58 

68 

78 

88 

98 

9 

19 

29 

39 

49 

59 

69 

79 

89 

99 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

It  is  im 

iportant  that  only 

such  numbers  as  the  cl 

ass  know  be 

placed  before  them  in 

the  above  table. 

Similarly  teach 

the  numbers 

to  1000.'  See  Speer's 

Elementary 

Arithmetic; 

p.  125 

III.   READING  AND  WRITING  NUMBERS  ABOVE  1000. 

1.  Instead  of  building  up  the  numbers  in  regular  order  above 
1000,  begin  by  teaching  the  reading  of  such  numbers  as  371,371  and 
follow  this  by  the  writing  of  such  numbers. 

2.  Next  teach  reading  followed  by  writing  such  numbers  as 
235,476;  46,840;  2,896.  Emphasize  that  the  three  right  hand  digits 
represent  units  and  the  next  three  digits  to  the  left  represent  thou- 
sands. Place  the  word  thousand  above  the  second  period  and  units 
above  the  first  period  and  have  the  number  read  for  a  time  as  follows: 
235  thousand,  476  units,  soon  dropping  the  word  'units". 

3.  Teach  such  numbers  as  234,054;  5,023;  87,004;  at  first  reading,' 
and  dictating  as  follows:  5  thousand,  no  hundred  23;  87  thousand, 
no  hundred  no  tens  4;  thus  emphasizing  tbe  ciphers  and  the  fact  that 
there  must  be  three  places  to  the  right  of  each  comma.  Frequent 
drill  in  wi*iting  and  reading  such  numbers  will  be  necessary  for  sev 
eral  terms  after  the  work  is  first  taught. 

4.  Teach  millions  as  above. 

19 


After  a  little  practice  of  this  sort,  dictate  in  the  usual  way,  at 
first  questioning  as  to  what  should  be  placed  in  hundreds  place,  etc. 

IV.  ADDITION. 

In  arriving  at  and  learning  the  addition  facts  the  child  should 
under  no  circumstances  be  allowed  to  count,  therefore  do  not  use 
single  objects  in  teaching  these  facts.  Some  teachers  will  cry  out 
against  this  as  a  great  heresy  in  that  it  violates  the  principle  of  ob 
jective  teaching.  This,  however,  is  an  error,  but  even  though  it  were 
true,  the  evils  resulting  from  the  objective  method  in  this  particular 
instance  far  outweigh  its  benefits  if  there  be  any  benefits.  The  child 
ivho  learns  his  facts  by  counting,  generally  never  learns  his  facts, 
paradoxical  as  that  may  sound.  When  he  has  a  column  of  figures  to 
add,  he  continues  to  count  through  life.  This  is  a  most  pernicious 
habit  and  results  in  slow  and  inaccurate  computation. 

However,  there  is  a  means  of  discovering  a  fact  without  develop- 
ing the  pernicious  counting  habit.  It  is  an  adaptation  of  the  method 
used  by  Montessori.  Have  wooden  rods  of  1  inch,  2  inches,  etc.,  up  to 
18  inches  in  length  with  the  length  written  upon  them  in  figures. 
The  rods  should  not  be  marked  off  in  inch  lengths  as  this  would,  as  in 
the  case  of  single  objects,  lead  to  counting.  To  teach  the  sum  of  two 
numbers,  as  2  plus  3,  have  each  child  find  a  two  inch  and  a  three  inch 
rod  and  place  them  end  to  end  and  then  find  another  rod  equal  in 
length  to  the  sum  of  the  two  rods. 

2.  Having  thus  discovered  that  2  plus  3  equals  5,  approach  all 
the  cortical  areas  of  th«  brain  as  indicated  in  the  discussion  regarding 
the  teaching  of  new  number  facts.  Teacher  repeat,  children  repeat, 
teacher  write  on  the  board,  children  visualize,  children  copy.  Many 
children  have  to  repeat  the  facts  before  they  can  give  the  results. 
They  should  be  drilled  to  give  answers  at  sight  or  sound  as  well  as 
by  repetition. 

3.  Teach  both  2  +  3  and  3  +  2  at  the  same  time,  as  this  is  but  ono 
fact. 

4.  Teach  both  horizontally  and  in  columns. 

5.  Do  not  try  to  teach  any  new  facts  till  all  the  old  facts  hav« 
been  thoroughly  learned. 

6.  The  order  of  teaching  facts  should  be  determined  by  sums 
rather  than  by  tables.  Teach  two  plus  two,  then  all  the  facts  whose 
sum  is  five,  then  those  whose  sum  is  six,  etc. 

7.  Preparation  for  subtraction  should  go  hand  in  hand  wit*'  the 
learning  of  the  addition  facts.     See  discussion  of  subtraction. 

8.  After  a  few  facts  have  been  learned,   column  addition   with 
sum  not  greater  than  nine  should  be  begun.     Also  simple  oral  exam- 
ples such  as  2  plus  2  plus  2.     During  the  second  half  year  column 
addition  should  be  continued  to  sum  18,     but  in  no     case  should  the 
column  exceed  the  number  facts  already  learned.     For  example  in  the 

20 


column  3-6-6,  if  the  class  know  the  facts  only  so  far  as  sum  15,  they 
could  add  downward  but  not  upward;  as  in  one  case  the  greatest  fact 
is  9  plus  6  which  they  have  learned,  while  in  the  other  case  the  fact 
is  12  plus  S  which  is  not  even  one  of  the  49  elementary  number  facts, 
which  include  only  the  facts  from  1  plus  1  to  9  plus  9.  In  column 
addition  do  not  point  or  allow  children  to  point.     This  is  important. 

9.  Carrying  may  alse  be  taught  during  the  second  term,  the  New 
York  State  Syllabus  advising  no  explanation.  Always  add  the  carry 
immediately.  Do  not  write  «t  at  the  top  of  the  next  column,  but  add 
it  to  the  top  digit  at  once  if  adding  down,  and  to  the  bottom  digit  if 
adding  up. 

10.  Each  day  the  previously  learned  facts  should  be  reviewed  by 
various  means:  by  placing  the  facfs  on  the  board,  the  child  to  fill  in 
the  results  orally  or  on  the  board,  or  by  copying  on  paper;  by  placing 
a  number  in  a  hollow  square  or  within  a  circle  of  numbers  and  adding 
it  in  turn  to  each  of  the  outer  numbers  without  any  pointing;  by  using 
number  cards;  by  means  of  ladders,  steps,  guessing  games,  and 
other  devices  that  may  occur  to  the  teacher. 

As  many  of  the  class  as  possible  should  try  to  reach  the  top  of 
ladder  or  steps  without  missing  a  step.  The  number  cards  should 
have  on  one  side  the  reverse  of  the  combination  found  on  the  other 
side;  that  is,  if  4  over  8  is  on  one  side,  8  over  4  should  be  on  the 
other  side.  Thus  the  teacher  can  see  the  fact  on  the  back  of  the  card 
as  she  brings  it  from  the  back  of  the  pack  to  the  front. .  This  is  im- 
portant. She  should  stand  where  all  the  class  can  see  the  cards  with- 
out effort  and  should  manipulate  them  rapidly  without  fumbling.  She 
should  vary  the  method  of  recitation;  sometimes  calling  on  the  chil- 
dren by  name  for  successive  facts,  rarely  calling  on  them  in  order, 
generally  having  one  child  rise  and  recite  several  facts  before  calling 
on  another  pupil.  When  one  child  misses,  say  "class"'  or  "tell',  and 
the  child  missing  should  pass  to  the  board  and  write  the  fact  in  the 
four  ways  for  writing  each  new  fact.  Children  should  recite  these 
facts  rapidly.  DO  NOT  GIVE  THEM  TIME  TO  OBTAIN  THE 
RESULT  BY  COUNTING.  Another  device  is  to  place  the  missed 
fact  back  in  the  pack  where  it  will  occur  again  soon  and  thus  give  the 
child  another  chance  to  recognize  the  fact.  Another  device  is  to  hand 
the  card  to  the  child  to  study.  Very  little  time,  perhaps  from  three 
to  five  minutes  a  day,  should  be  spent  in  this  drill. 

In  the  guessing  game  the  teacher  or  one  of  the  pupils  says:  "I 
am  thinking  of  two  numbers  whose  sum  is  14."  The  class  then 
guess  as  called  upon  by  the  teacher,  "Is  it  7  and  7  are  14?"  "Is  it  8 
and  6  are  14?"  etc.,  till  the  correct  combination  is  guessed.  It  is 
generally  well  for  the  child  to  tell  the  teacher  what  fact  he  has 
chosen.     As  games  take  time,  they  should  be  used  with  discretion. 

After  carrying  has  been  taught,  see  that  all   examples  given  be- 
fore beginning  series  illustrate  both  carrying  and  not  carrying,  that 

21 


the  children  may  constantly  be  compelled  to  judge  whether  they  must 

carry  or  not. 

25224  74476  34332  63648  34716 

64454  21320  24023  34231  53243 

23234  43553  85579  71587  98929 

Have  class  rise,  add  down,  and  be  seated  as  fast  as  the  result  of 
a  column  is  found.  Calling  on  some  child,  often  the  one  seated  last, 
ask  the  result.  The  child  may  answer  in  the  first  example,  second 
column,  "ten".  As  if  the  teacher  considered  this  result  correct,  she 
should  ask  how  many  have  ten.  Why  not,  Charles?  Because  there 
i.s  one  to  carry.  Add  it  again  orally,  Frank.  See  that  the  carry  is 
made  at  once  whether  beginning  with  the  bottom  or  top  figure.  This 
cannot  be  over  emphasized.  In  the  fourth  column,  Frank  may  this 
time  give  the  result  as  thirteen.  Question  as  before,  bringing  out 
that  there  is  no  carry  this  time.  This  method  will  keep  the  children 
constantly  on  the  alert  to  see  whether  there  is  a  carry.  After  a  few 
minutes  of  this  practice,  send  the  class  to  the  board  to  work  examples 
already  on  the  board.  As  fast  as  two  children  have  their  results 
correct,  have  them  erase  results  and  exchange  examples,  thus  keep- 
ing all  busy. 

One  point  of  importance  that  all  teachers  should  observe  is  to 
have  chalk  and  erasers  ready  at  the  board  before  class  that  no  time 
need  be  wasted  in  class. 

See  Brown  and  Coffman,  op.  cit.,  pp.  150-159;  Bailey,  op.  cit., 
Lesson  16;  Walsh,  Methods  in  Arithmetic,  Chap.  II;  McLellan  and 
Dewey,  pp.  195-200;  Smith,  pp.  114-117;  McMurry,  pp.  30-53. 

V.  SERIES  AND  COLUMN  ADDITION. 

There  are  three  classes  of  series: 

1.  Sum  of  right  hand  digits  less  than  ten.  Illustration:  sum  of 
right  hand  digits  7. 

24     44     84     64     73     33     53     94     14 
333344433     etc. 

2.  Sum  of  right  hand  digits  ten. 
28  plus  2,  38  plus  2,  42  plus  8,  etc. 

3.  Sum  of  right  hand  digits  more  than  ten.  Illustration,  sum 
of  right  hand  figures  12: 

27  plus  5,  77  plus  5,  85  plus  7,  etc. 

The  teacher  may  give  the  series  orally,  the  children  giving  the 
results;  the  series  may  be  placed  vertically  on  the  board  as  above;  or 
the  single  digit  number  may  be  placed  in  the  center  of  a  series  of  two 
digit  numbers,  all  having  the  same  right  hand  digit,  the  child  giving 
results  in  regular  order  from  right  to  left  or  left  to  right,  no  pointing. 
In  the  oral  work  the  teacher  should  say  27-5.  If  the  child  hesitates, 
the  teacher  should  say  7-5,  and  the  child  should  say  32.     If  he  says  12, 

22 


the  teacher  should  repeat  27-5.  If  the  child  still  hesitates,  the  teach- 
er should  return  to  7-5,  and  on  receiving  the  reply  12  again,  the 
teacher  should  say,  what  is  the  right  hand  figure?  Then  what  is  the 
right  hand  figure  in  the  sum  of  27  and  5?  27-5?  If  the  child  still 
cannot  give  the  result  place  the  numbers  vertically  on  the  board  and 
add  as  in  column  addition. 

One  purpose  of  series  drill  is  preparation  for  column  addition. 
The  two  should  therefore  go  hand  in  hand.  Children  who  may  know 
readily  that  7  plus  5  is  12,  may  not  know  without  counting  that  27 
plus  5  is  32.  In  the  column  example  6-8-3  the  child  may  start  cor- 
rectly by  saying  14  (he  should  never  say  6  plus  8  are  14,  or  even  6-14, 
but  only  14)  and  then,  though  he  may  know  that  4  and  3  are  7,  he 
may  not  know  that  14  and  3  are  17.  The  teacher  should  say  "4-3". 
If  the  child  says  "17",  do  not  compel  him  to  say  "7",  as  17  was  the 
answer  wanted.  If  he  says  "7",  you  may  be  compelled  to  say,  "Then 
how  much  is  14  and  3"?  If  the  series  4-3  is  on  the  board  at  the  time, 
the  teacher  may  merely  refer  to  this. 

During  the  third  half  year,  series,  sum  less  than  10  should  begin. 
Such  work  should  at  first  be  oral,  followed  with  blackboard  work  sim- 
ilar to  the  illustrations  given  above,  and  accompanied  by  appropriate 
column  addition.  The  work  should  continue  as  necessary  through  the 
following  grades. 

The  series  should  be  built  up  something  as  follows:  0  plus  1  to 
0  plus  9;  1  plus  1;  1  plus  2;  etc.  to  1  plus  9;  miscellaneous  practice. 
The  child's  readiness  in  the  miscellaneous  practice  and  column  addi- 
tion will  show  how  successful  the  series  has  been.  Follow  this  with 
2  plus  2;  miscellaneous  practice  applying  all  series  hitherto  drilled 
on;  2  plus  3;  miscellaneous;  3  plus  3;  miscellaneous,  etc.,  up  to  4  plus 
5. 

As  soon  as  scries  2  plus  2  has  been  learned,  such  examples  as 

the  following  should  be  given.   The  children  should  add  from  the  top 

down  as  follows:        12-14-15;       5-7-12-14;        9-12-13-14. 

8  4  6  The  addition  should  be  oral  that  the  teacher  my  know 

3  2  6  how  the  pupil  is  arriving  at  the  result.     No  other  num- 

15  2  bers  or  words  than  those  used  above  should  be  used  by 

12  1  the  children  in  adding.     If  the  examples  are  added  from 

the  bottom  up,  they  will  not  be  in  accord  with  the  series. 

FIRST  SERIES— SUM  OF  RIGHT  HAND  DIGITS  LESS 

THAN  TEN. 

The  following  examples  are  constructed  progressively  according 
to  the  series: 

347  245  274  789  456  264  639  3  57 
653  755  726  211  544  736  361  643 
876  318  539  423  543  589  986  518 


23 


243 

148 

431 

142 

246 

348 

768 

648 

867 

962 

679 

899 

854 

853 

343 

462 

649 

871 

572 

5X  6 

765 

175 

678 

631 

110 

16 

1  10 

111 

111 

1  10 

110 

118 

579  594  908  870 

643  426  294  948 

222  222  272  124 

103  212  111  12 


876 

543 

375 

294 

206 

458 

902 

980 

456 

777 

777 

938 

885 

884 

399 

339 

223 

233 

213 

322 

141 

213 

231 

222 

10  1 

112 

211 

110 

323 

111 

123 

117 

987 

876 

567 

452 

378 

597 

397 

958 

3  5  7 

555 

777 

777 

666 

645 

824 

463 

122 

211 

122 

413 

222 

424 

44  1 

24  1 

21  1 

124 

211 

124 

211 

100 

1  14 

1  14 

767  898  858  375 

574  225  172  848 

322  243  333  413 

113  311  313  131 


397 

894 

836 

975 

789 

597 

34  5 

738 

823 

629 

696 

54  7 

743 

655 

667 

589 

532 

232 

225 

235 

235 

423 

245 

331 

125 

122 

121 

121 

111 

1  12 

521 

121 

489 

638 

543 

178 

837 

9  "  7 

839 

847 

954 

696 

779 

977 

586 

585 

694 

598 

334 

323 

225 

322 

224 

235 

214 

322 

1  10 

101 

231 

301 

121 

101 

13  1 

1  10 

876  593  956  147 

567  749  568  996 

434  242  224  341 

110  214  141  414 


849 

849 

748 

847 

958 

678 

567 

596 

773 

273 

498 

389 

264 

655 

6  '  5 

466 

264 

146 

422 

442 

656 

323 

604 

224 

112 

621 

221 

1  10 

10  1 

222 

162 

6  12 

24 


847 

876 

638 

938 

538 

678 

587 

596 

683 

349 

677 

294 

746 

746 

765 

847 

353 

313 

223 

153 

57  1 

44  1 

53  1 

245 

1  15 

35  1 

351 

5  13 

133 

123 

1  1  3 

2  1  1 

847 

948 

657 

847 

673 

537 

586 

5  09 

465 

484 

746 

674 

668 

777 

636 

894 

447 

237 

324 

301 

531 

513 

473 

454 

230 

220 

272 

177 

127 

172 

203 

132 

976 

849 

1  73 

438 

838 

489 

473 

821 

678 

573 

987 

974 

786 

856 

792 

E  99 

312 

216 

433 

225 

214 

342 

428 

243 

33 

361 

306 

362 

161 

312 

306 

326 

678 

678 

483 

459 

543 

678 

594 

73  1 

954 

542 

534 

497 

537 

853 

4  "  0 

947 

121 

434 

438 

432 

464 

434 

429 

286 

245 

245 

543 

501 

454 

24 

523 

34 

SECOND  SERIES— SUM  OF  RIGHT  HAND  DIGITS  TEN. 

At  first  the  class  should  find  any  two  successive  digits  whose  sum 
is  ten  and  the  teacher  or  child  should  connect  them  with  a  curve. 
Then  for  a  time  the  tens  should  be  found  before  adding,  but  the 
rurves  should  be  dropped.  As  soon  as  possible  the  child  should  ac- 
quire the  ability  of  finding  the  combinations  of  ten  as  he  adds.  Note 
that  in  these  examples  the  passage  from  the  teens  to  the  twenties 
and  from  the  twenties  to  the  thirties,  etc.,  is  made  by  a  combination 
of  10.  Before  beginning  the  column  addition,  give  series  drill  as  be- 
fore on  the  combination  to  be  introduced  in  the  example. 

The  child  should  add  rapidly  as  follows  (see  first  two  examples) : 
15-20-23-33-37;  9-15-20-26-36-38;  12-22-25-30-38;  11-15-20-24-34-39; 
11-21-25-30-38.  13-15-20-29;  12-15-20-27-37;  11-21-25-30-38;  9-15- 
20-24-34-37;     11-15-20-28. 


88968 

46853 

58947 

78948 

29498 

54567 

46555 

85389 

65698 

14969 

55555 

4  "535 

65464 

65445 

76743 

44363 

54252 

42444 

45345 

35373 

55555 

25275 

44444 

44454 

37373 

75455 

25554 

46646 

32454 

33577 

15434. 

43855 

35666 

57365 

43437 

25 


49935 

94886 

95498 

98657 

85899 

96778 

67386 

78888 

69894 

89279 

34773 

63788 

24282 

15219 

29612 

37577 

42878 

28272 

86119 

91981 

56567 

24836 

88738 

84778 

72727 

57344 

54724 

82822 

24612 

95393 

73934 

36372 

52297 

21491 

13179 

THIRD  SERIES— SUM  OF  RIGHT  HAND  DIGITS 
GREATER  THAN  TEN. 

First  give  series  and  examples  in  which  nine  is  added  to  various 
numbers,  calling  attention  to  the  fact  that  the  right  hand  digit  of  the 
result  is  one  less  than,  and  the  left  hand  digit  one  greater  than  the 
corresponding  digits  of  the  number  added  to  nine.  Then  give  the 
series  in  the  order  illustrated  in  the  following  examples: 


98788 

48398 

91398 

99799 

24898 

79998 

84638 

28312 

99789 

68688 

35989 

56435 

42777 

12893 

46656 

99875 

44574 

67464 

82286 

73466 

67998 

47757 

75669 

78336 

33347 

93949 

63763 

96447 

78844 

64563 

29489 

66326 

94744 

37044 

76663 

42348 

89299 

78989 

39769 

99248 

72689 

75859 

93948 

88349 

88278 

78465 

59874 

76846 

28885 

47832 

57558 

71635 

64765 

55898 

86866 

79385 

34646 

87677 

18684 

83467 

34627 

89494 

65183 

88688 

26545 

57656 

64475 

64874 

87858 

88888 

43438 

99438 

48768 

45766 

62342 

88344 

68687 

98888 

85455 

86454 

68787 

58888 

77878 

88788 

VI.  SUBTRACTION. 

Teach  subtraction  by  the  Austrian  or  addition  process.  Tne 
following  preparation  should  be  given  while  teaching  the  addition 
facts:  6-f-?=9;  3+?=  9.  Arrange  on  board  or  cards  both  horizon- 
tally and  vertically. 

Bring  out  the  subtraction  idea  concretely.  If  you  had  4  apples 
and  gave  me  2,  how  many  would  you  have  left?     How  many  Ho  you 

26 


have  to  add  to  2  to  have  4?  Then  4 — 2=?  If  you  have  5  marbles 
and  lose  2,  how  many  do  you  have  left?  How  many  do  you  add  to  2 
to  have  5?     Then  5—2=? 

Give  several  examples  like  the  above  that  have  come  within  the 
actual  experience  of  the  child  and  bring  out  that  we  may  find  the  re- 
sult of  subtraction  by  adding  to  the  subtrahend  a  number  that  will 
give  the  minuend.  Make  constant  use  of  the  terms  subtraction,  minu- 
end, subtrahend,  and  difference,  that  the  children  may  become  thor- 
oughly familiar  with  their  meaning.  It  is  much  better  to  be  able  to 
use  the  terms  accurately  without  hesitation  than  to  learn  the  defini- 
tion without  making  use  of  the  term  itself  in  the  work.  Thus,  in  fact, 
the  child  has  learned  the  content  of  most  of  the  words  he  uses. 
Comparatively  few  words  have  been  actually  defined  for  him. 

For  practice  give  exercises  as  follows: 

1.  Each  digit  in  the  subtrahend  equal  to,  or  less  than,  the  cor- 
responding digit  in  the  minuend. 

47948635867208 
—  1523061  2342104 

Emphasize  that  we  add  down  in  subtraction;  that  is,  we  add  the 
lower  number,  or  the  subtrahend,  to  the  number  that  we  place  below, 
or  the  difference,  in  order  to  give  the  minuend,  or  the  upper  number. 
At  first  the  teacher  will  have  to  question  the  child  as  to  how  much 
must  be  added  to  4  to  give  8,  etc.  Then  the  child  should  ask  himself 
these  questions,  but  as  soon  as  possible  he  should  add  as  follows  with- 
out any  extra  words:  4  plus  4  is  8;  0  plus  0  is  0;  1  plus  1  is  2;  2  plus 
5  is  7;  etc.;  putting  down  result  as  he  repeats  it.  Even  this  should  be 
abandoned  and  the  result  seen  at  a  glance  and  put  down  without  a 
word,     Not  much  time  should  be  spent  on  this  step. 

2.  Digits  in  the  minuend  less  than  the  corresponding  digits  in 
the  subtrahend.  If  the  results  of  the  first  exercises  are  written  be- 
fore the  subtraction  begins,  the  method  will  be  seen  more  clearly. 

734695037826 
—  17892  3  613917 


55  5  771423909 

As  in  the  case  of  smaller  numbers,  we  add  the  subtrahend  and 
difference  to  obtain  the  minuend.  Thus:  7  plus  9  is  16.  In  addition 
we  should  put  down  the  6  and  carry  the  1,  but  the  6  is  already  written 
in  the  minuend,  so  all  we  have  to  do  after  setting  down  the  9  is  to 
carry  the  1  exactly  as  we  do  in  addition.  Adding  the  1  that  we  carry 
to  the  1  in  the  subtrahend  gives  us  2  and  we  say,  2  plus  0  is  2.  Just 
as  in  addition  we  add  the  number  carried  to  the  top  addend  without 
any  extra  words,  merely  giving  the  result  of  such  addition,  so  here 
we  merely  add  the  amount  to  be  carried  to  the  subtrahend,  which  in 
this  case  is  equivalent  to  the  top  addend.     We  then  add  this  result  to 

27 


the  difference,  which  is  the  equivalent  to  the  next  addend.  Thus,  9 
plus  9  is  18;  4  plus  3  is  7;  1  plus  2  is  3;  6  plus  4  is  10;  4  plus  1  is  5,  2 
plus  7  is  9;  9  plus  7  is  16;  9  plus  5  is  14;  8  plus  5  is  13;  2  plus  5  is  7. 

Call  attention  to  the  fact  that  when  the  digit  in  the  subtrahend 
is  greater  than  that  in  the  minuend,  we  add  enough  to  give  a  number 
whose  right  hand  digit  is  the  digit  in  the  minuend.  That  is,  we  can- 
not add  to  7  to  give  6  therefore  we  add  enough  to  give  16.  When  we 
do  this,  we  must  carry  the  1.  After  going  over  several  examples  that 
the  teacher  has  solved  on  the  board,  place  on  the  board  exercises  to 
be  solved. 

54892070318 
—  17925375236 

The  first  2  will  be  given  without  difficulty.  Teacher  question  as 
follows:  Can  we  add  to  3  to  give  1?  We  must  add  enough  to  give 
how  much?  Ans.  11.  How  much  must  we  add  to  3  to  give  11? 
Teacher  or  child  sets  down  8.  As  the  figure  in  the  minuend  is  only  1 
instead  of  11,  what  shall  we  do?  Ans.  Carry  1.  To  what  shall  we 
add  it?  Then  3  and  how  many  make  3?  How  many  to  carry?  Why 
not?  What  shall  we  say  next?  Ans.  5  plus  9  is  14.  Is  there  any- 
thing to  carry?  Etc.  After  working  several  thus  with  the  class, 
send  individuals  to  the  board  to  work  under  the  direction  of  the  teach- 
er with  suggestions  by  the  rest  of  the  class;  then  send  the  whole  class 
to  the  board,  using  the  "chalk  and  talk"  method.  Be  sure  that  the 
voice  does  not  indicate  the  answer  to  the  question  whether  there  is 
anything  to  carry. 

From  now  on  every  example  should  illustrate  both  carrying  and 
not  carrying  in  irregular  order,  that  the  child  may  be  compelled  con- 
stantly to  determine  whether  there  is  a  carry  or  not. 

As  soon  as  possible  the  class  should  add  rapidly  as  follows: 
G  plus  2  is  8;  3  plus  8  is  11;  3  plus  0  is  3;  5  plus  9  is  14;  8  plus  9  is  17; 
4  plus  6  is  10;  etc.     At  length  the  result  should  be  seen  at  a  glance. 

The  advantage  of  the  addition  method  is  that  only  one  set  of 
facts  has  to  be  learned  instead  of  two.  If  the  teacher  is  not  watchful, 
however,  one  drawback  will  be  found:  in  the  solution  of  problems 
requiring  subtraction,  the  subtraction  idea  will  be  lost  sight  of.  The 
teacher  must  emphasize  the  fact  that  subtraction  means  take  away 
and  that  we  merely  find  how  much  to  take  away  by  means  of  the 
addition  facts.  Simple  problems  in  subtraction  should  be  given  from 
the  first  to  obviate  this  difficulty. 

See  Brown  and  Coffman,  pp.  160-163;  Walsh,  Chap.  Ill;  Bailey, 
Lesson  17;  Smith,  pp.  121-122;  McLellan  and  Dewey,  pp.  200-206. 

VII.  MULTIPLICATION. 

No  attempt  to  teach  multiplication  should  be  made  until  all  the 
addition  facts  have  been  thoroughly  learned.     Teaching  the  four  fun- 

28 


damental  operations  simultaneously  is  the  great  weakness  as  well  as 
the  chief  feature  of  the  Grube  method. 

McMurry  (p.  54)  suggests  the  following  order  of  teaching  the 
multiplication  tables:  "10s,  2s,  5s,  4s,  8s,  3s,  6s,  9s,  7s."  This  is  an 
improvement  over  the  old  order. 

In  an  excellent  little  monograph  on  teaching  "The  multiplication 
tables",  Gildemeister  uses  the  following  order:  2s,  Is,  10s,  lis,  9s, 
6s,  0s,  3s,  4s,  6s,  7s,  8s,  12s.  A  study  of  this  pamphlet  will  repay  any 
teacher  of  multiplication,  though  it  proposes  a  too  early  introduction 
of  division. 

At  first  the  facts  should  be  found  inductively.  To  teach  2x3  =  6, 
give  3  plus  3  is  6,  that  is,  two  3s  are  6,  or  2  times  3  is  6.  2  plus  2 
plus  2  is  6;  that  is,  three  2s  are  6,  or  3  times  2  is  6.  Teach  2x3  and 
3x2  as  one  fact,  writing  both  horizontally  and  vertically.  See  p.  6 
for  method  of  teaching  a  new  number  fact. 

Concentrate  upon  one  fact  till  that  is  known.  Sooner  or  later 
the  child  must  learn  each  fact  as  an  individual  fact,  or  he  will  not 
'earn  it  at  all.  As  in  addition,  drill  by  means  of  oral  questions,  cards, 
fi;cts  on  board  or  paper,  circle,  etc.  Bring  together  such  facts  as 
3x4  and  2x6.  Give  table  of  squares  as  2x2,  3x3,  etc.  Prepara- 
tion for  division,  as  3x?=6,  should  go  hand  in  hand  with  multipli- 
cation. 

When  assigned  facts  have  been  written  out  by  the  pupil,  a  chart 
of  the  multiplication  tables  may  be  placed  before  him  that  he  may 
compare  his  results  with  it  to  see  whether  his  are  correct.  He  may 
thus  make  his  own  corrections  and  study  upon  those  missed,  or  skip- 
ped because  he  was  in  doubt  about  them.  The  experience  of  many 
teachers  has  been  that  when  the  only  means  taken  to  teach  the  table? 
is;  by  hanging  the  chart  before  the  class  while  examples  are  to  be 
worked,  children  generally  depend  entirely  upon  the  chart  and  do  not 
commit  the  facts.  As  indicated  above,  the  facts  should  first  be  writ- 
ten and  the  chart  merely  used  for  correction. 

As  soon  as  a  few  facts  have  been  learned,  they  should  be  used  in 
examples.  The  teacher  should  at  first  place  the  examples  on  the 
board  and  write  the  results  as  the  children  give  them.  As  soon  as 
2  X  3  is  reached,  the  following  examp'e  may  be  used:  2x213.  Send 
class  to  board.  Time  may  be  saved  by  multiplying  the  result  of  one 
multiplication  by  another  multiplier. 

As  carrying  has  already  been  taught  in  addition,  the  idea  ought 
to  present  little  difficulty  here.  As  an  aid  to  carrying,  daily  drill 
should  be  given  with  such  exercises  as  4x7  +  5;  7x8  +  6.  This  drill 
should  continue  till  the  most  difficult  operations  in  multiplication  can 
be  performed  with  facility. 

See  that  the  children  use  readily  the  terms  multiplication,  multi- 
plier, multiplicand,  and  product.  From  the  beginning  give  simple 
problems  applying  the  principle  of  multiplication.     At  first  oral  prob- 

29 


lems  should  be  given.  When  written  problems  of  any  type  are  intro- 
duced, the  numbers  used  should  be  as  small  as  those  used  in  the  oral 
problems.  The  child  can  thus  give  his  entire  attention  to  the  new 
difficulty  of  expressing  in  writing  what  he  has  heretofore  expressed 
orally,  without  being  distracted  by  the  size  of  the  numbers  which  may 
be  gradually  increased  after  he  has  become  familiar  with  the  method. 
That  the  first  written  solution  of  a  problem  and  the  use  of  large  num- 
bers in  solutions  are  both  serious  difficulties  will  be  the  testimony  of 
any  teacher  of  experience.  Adhere  rigidly  to  the  pedagogical  princi- 
ple to  introduce  but  one  difficulty  at  a  time.  The  violation  of  this  rule 
inevitably  leads  to  confusion.  In  the  working  of  problems  emphasize 
that  the  product  is  always  the  same  in  kind  as  the  multiplicand.  If 
we  multiply  apples,  the  result  is  apples;  if  we  multiply  marbles,  the 
result  is  marbles;  if  we  multiply  an  abstract  number,  the  result  is  an 
abstract  number.  2  x  $3  should  be  read  2  times  $3.  $3x2  should  be 
read  $3  multiplied  by  2. 

Teach  early  multiplication  by  10  by  merely  annexing  a  cipher  to 
the  multiplicand.  Give  much  practice  in  applying  this  short  process 
and  allow  no  other.  Later  give  similar  practice  in  multiplying  by 
powers  of  10.  Never  allow  a  child  to  multiply  through  by  zero  and 
place  a  row  of  zeroes  as  a  partial  product. 

Teach  multiplication  by  the  factors  of  a  number  in  varied  order. 
30  X  $25  =  2  x  3  X  5  X  $25  =  3  X  5  X  2  X  $25  =  2  X  ~  X  3  X  $25. 

See  Brown  and  Coffman,  pp.  160-167;  Bailey,  Lesson  18;  Walsh, 
Chap.  IV;  McLellan  and  Dewey,  pp.  207-220. 

VIII.  DIVISION. 

Division  and  Partition.  Whether  or  not  the  terms  division  and 
partition  are  both  employed,  the  distinction  in  ideas  implied  should  be 
made.  That  is,  in  multiplication  the  product  is  the  result  of  multi- 
plying together  two  factors,  one  of  which  may  be  concrete  an  1  at 
least  one  abstract.  When  the  process  is  reversed  in  division  the  pro- 
duct becomes  the  dividend,  one  of  the  factors  the  divisor,  and  the 
other  factor  the  quotient.  If  $15  is  the  product  of  3  and  of  $5,  then 
$15  may  be  divided  by  $5  or  by  3.  $15  divided  by  $5  equals  3,  ard 
$15  divided  by  3  equals  $5.  Much  practice  should  be  given  in  deter- 
mining whether  the  quotient  is  abstract  or  concrete. 

After  division  is  well  in  hand,  it  is  well  to  check  examples  by 
multiplying  the  quotient  by  the  divisor.  Multiplication  examples  also 
should  be  checked  by  dividing  the  product  by  the  multiplier,  not  by 
the  multiplicand  as  the  child  may  then  merely  copy  the  mistakes  he 
may  have  made  in  his  multiplication. 

For  checks  of  various  kinds,  see  Brown  and  Coffman,  Chap.  IV. 

Simple  division  problems  should  be  used  from  the  first.  Also 
use  and  have  children  use  terms  division,  dividend,  divisor,  quotient, 
remainder,  and  trial  divisor. 

30 


Teach  division  as  the  reverse  of  multiplication.  As  a  prepara- 
tion while  teaching  the  multiplication  tables  have  the  child  for  ex- 
ample tell  by  what  we  must  multiply  7  to  give  28.  7x  ?  =28.  4  X  ? 
—  28.  Then  when  we  come  to  division,  we  refer  to  this  and  bring  out 
that  28  divided  by  4  gives  7.  As  in  subtraction,  so  here  we  must 
guard  against  the  danger  of  not  grasping  the  division  idea.  To  ob- 
viate this  difficulty,  employ  many  simple  problems  within  the  experi- 
ence and  grasp  of  the  child. 

In  division  as  in  addition  it  is  of  the  greatest  importance  to  take 
one  step  at  a  time..  The  following  series  of  progressive  steps  has 
worked  itself  out  as  the  result  of  class  room  observation. 

IX.  SHORT  DIVISION. 

In  both  short  and  long  division,  place  the  quotient  above  the  div- 
idend. The  first  quotient  figure  should  be  directly  above  the  right 
hand  figure  of  the  first  partial  dividend.  Each  succeeding  figure  in 
the  dividend  should  have  a  corresponding  figure  above  it  in  the  quo- 
tient.     Insist  on  exact  placing  of  quotient  figures. 

First  teach  division  within  the  multiplication  tables,  (a)  Divi- 
sor exactly  contained  in  the  dividend;  as,  7)14,  7)21,  etc.  (b)  Divi- 
sion with  a  remainder;  as,  7)16,  7)69.  etc.  Before  giving  more  diffi- 
cult examples,  give  thorough  drill  in  dividing  any  number  below  20 
by  2,  any  number  below  30  by  3,  etc.,  up  to  89  divided  by  9.  Write 
the  examples  on  board  or  give  them  orally,  having  child  give  quotient 
and  remainder  orally,  or  send  class  to  the  board  to  work  several  ex- 
amples as  rapidly  as  possible.  The  teacher  may  place  several  exam- 
ples at  each  child's  place  before  class,  or  she  may  place  them  on  some 
conspicuous  board  or  chart,  numbering  them  in  regular  order.  Num- 
ber the  children  in  irregular  order  to  prevent  copying;  as,  1,  4,  7,  10, 
2,  5,  8,  11,  3,  6,  9,  12.  Have  each  child  begin  with  the  example  cor- 
responding with  his  number,  and  as  soon  as  he  works  one  example, 
begin  the  next  one  without  waiting.  Those  beginning  with  the  higher 
numbers  should  turn  back  to  example  1  as  soon  as  they  finish  the  last 
one  given.  The  teacher  should  pass  around  giving  a  "C"  for  each 
correct,  and  a  "0"  for  each  incorrect  result.  An  honor  roll  may  be 
kept  of  those  working  a  certain  number  without  error.  Another  de- 
vice is  to  place  the  divisor  within  a  circle  of  dividends.  This  device 
should  be  continued  for  rapid  drill  even  after  formal  work  in  short 
division  begins. 

After  division  within  the  tables  is  well  in  hand,  give  thorough 
drill  with  examples  of  progressive  difficulty  as  follows:  4)4881. 
3)30639,  7)14707,  6)121824,  8)168056,  4)487,  6)105157,4)52804, 
3)741.  Give  easy  divisors  at  first,  gradually  increasing  their  difficulty. 
For  an  abundance  of  examples  illustrating  most  of  these  type  exam- 
ples, see  Woodficld,  A  manual  on  the  teaching  of  division. 

As  a  preparation  for  long  division,  teach  short  division  by  11 

31 


and  12.  Teach  division  by  10  by  merely  striking  off  the  right  hand 
figure,  the  left  hand  figures  being  the  quotient  and  the  right  hand 
figure  the  remainder. 

X.  LONG  DIVISION. 

Never  allow  long  division  with  divisors  less  than  11.  Place  the 
result  above  the  dividend. 

1.  Division  by  11  and  12.  As  a  preparation  use  short  division 
with  divisors  11  and  12.  Have  a  child  tell  how  he  worked  a  certain 
example  and  the  teacher  put  down  all  the  work  that  the  child  has 
done  mentally,  calling  attention  to  the  successive  steps.  After  work- 
ing several  examples  thus,  send  a  child  to  the  board  to  go  through 
the  written  steps  under  the  guidance  of  the  teacher.  Finally  send 
the  whole  class  to  the  board  and  use  the  "chalk  and  talk"  method. 
Suppose  the  example  to  be  12)  168.  How  many  times  is  12  contained 
in  16,  Mary?  Once.  Where  do  we  write  the  1?  Above  the  6.  Write 
it,  class.  1  times  12,  James?  Where  shall  we  write  the  12?  Write 
it,  class.  What  shall  we  do  next,  Anna?  Bring  down  the  8.  Class 
brings  down  the  8.  12  is  in  48,  Harold?  Etc.  Continue  this  process 
till  the  class  can  take  the  steps  unaided.  Use  examples  that  will  give 
but  two  places  in  the  result.  Continue  with  divisors  11  and  12  till  the 
class  is  familiar  with  the  order  of  the  steps:  1) divide,  2.)  multiply. 
3.)  subtract,  4.)  bring  down,  repeat. 

2.  Each  quotient  figure  found  by  dividing  the  first  digit  or  first 
two  digits  of  the  partial  dividend  by  the  first  digit  of  the  divisor, 
failed  the  trial  divisor.  (See  Shutts,  Handbook  of  Arithmetic,  p.  27). 
The  following  examples  illustrate  this  step:  Divisor  13,  numbers 
from  148  to  149,  156  to  159,  260-269,  273-9,  286-9;  divisor  14,  140-9, 
154-9,  168-9,  294-9;  divisor  21,  231-9,  441-9,  651-9,  861-9,  252-9,  462-9, 
672-9,  882-9,  273-9,  483-9,  504-9,  714-9,  903-9,  924-9;  22),  242-9, 
264-9,  286-9,  462-9,  484-9,  682-9;  23),  253-9,  276-9,  483-9,  506-9, 
713-9,  943-9,  966-9;  24),  264-9,  288-9;  31),  341-9,  651-9,  961-9, 
372-9,  682-9,  992-9,  401-9,  713-9,  961-9,  992-9,  32),  352-9,  672-9, 
384-9,  416-9,  704-9,  736-9;  33),  363-9,  396-9,  693-9,  726-9;  34). 
374-9,  680-9,  714-9;  35),  420-9,  735-9,  770-9;  36),  432-9,  756-9. 
792-9:  37),  404-9,  444-9,  777-9,  814-9;  41),  451-9,  861-9,  492-9, 
820-9:  42),  462-9,  882-9;  43),  473-9;  44),  484-9;  21),  1071-9, 
1260-9,  1281-9,  1470-9,  1491-9  1113-9,  1134-9,  1302-9,  1323-9,  1554-9, 
1743-9,  1932-9,  1974-9;  31),  1271-9,  1550-9,  1581-9,  1333-9,  1643-9, 
:922-9,  2232-9,  2573-9,  2883-9;  41),  1271-9,  1681-9,  1722-9,  2132-9, 
2296-9,  2337-9;  51),  1071-9,  1581-9;  52),  1092-9;  61),  1281-9, 
1891-9;  71),  1491-9.  For  further  examples  and  fuller  development 
of  this  topic,  see  Woodfield,  op.  cit.,  pp.  12-22. 

3.  Quotient  figure  one  less  than  the  number  of  times  the  trial 
divisor  is  contained  in  the  trial  dividend.  Here  emphasize  the  two 
principles:  (a)  The  subtrahend  should  not  be  greater  than  the  par- 
tial dividend.     If  it  is,  it  is  a  sign  that  the  quotient  figure  should  be 

32 


smaller,  (b)  The  remainder  should  not  be  greater  than,  or  equal  to, 
the  divisor.  If  it  is,  it  is  a  sign  that  the  quotient  figure  should  be 
larger.  Examples:  divisor  22),  400-417,  600-615,  800-813;  23), 
400-413,  600-619,  800-804,  810-827;  24),  400-419,  600-619,  800-815; 
25),  400-409,  600-619,  810-824;  26),  416-419,  600-619,  820-831;  32), 
1230-1247,  1504-1535,  1810-1823;  31),  1510-8,  1820-8;  33),  1200-1220, 
1230-1253,  1500-1517;  43),  1610-1633,  1650-1676;  44),  2120-2155, 
2920-2947;     46),  3960-3999;     47),  2980-3007;     54),  3000-3023. 

4.  Quotient  figure  more  than  one  less  than  the  number  of  time? 
the  trial  divisor  is  contained  in  the  trial  dividend.  Here  the  divisors 
from  14  to  19  should  be  used.  As  19  is  nearer  20  in  value  than  it  is 
to  10,  take  2  instead  of  1  as  the  trial  divisor.  With  divisors  14  to  18, 
take  both  1  and  2  as  trial  divisors  and  try  as  quotient  figure  some 
digit  between  the  two  results  of  trial.  Here  it  must  be  noted  that  in 
dividing  such  numbers  as  1288  by  14,  1  is  not  the  trial  dividend  as  12 
is  less  than  14.  In  such  cases,  as  9  is  the  largest  digit,  we  try  9  or 
some  smaller  number  as  the  trial  quotient. 

5.  More  than  two  figures  in  the  quotient.  Promiscuous  exam- 
ples with  two  figures  in  the  divisor  may  now  be  taken  up.  These  may 
be  found  in  any  text  book  or  taken  from  such  cards  as  Maxson's  self- 
keyed  number  cards. 

6.  More  than  two  figures  in  the  divisor.  In  dividing  by  such 
numbers  as  20,  300,  4000,  etc.,  do  not  allow  long  division.  Strike  off 
as  a  remainder  as  many  places  in  the  dividend  as  there  are  ciphers  at 
the  right  of  the  significant  figure  of  the  divisor,  and  divide  by  short 
division.  If  there  is  an  extra  remainder  after  dividing  by  the  digit, 
place  this  before  the  remainder  struck  off. 

124  178  8 


6/)  744/0  4/00)712/45  8/000)65/721—1721 

See  Brown  and  Coffman,   pp.   167-170;     Bailey.   Lessons   19,  20; 

McLellan  and   Dewey,    pp.    119-143,   220-240;     McMurray,   pp.    60-79; 

Walsh,  pp.  90-105. 

X.  ROMAN  NOTATION. 

Systematize  the  method  of  teaching  the  Roman  notation.     First 
teach  the  following: 

1=1,  10=X,  100=C,  1000  =  M. 

After  this  is  thoroughly  learned,  give  the  following: 
2=11.  20  =  XX,  200=CC,  2000  =  M\I. 

3=111,  30=XXX,  300=CCC,  3000  =  MMM. 

Next  give:  

5  =  V,  50  =  L,  500=D,  5000  =  V 

Follow  with:  

6=VI,  60  =  LX,  600  =  DC.  6000=V1 

7=VII,  70=LX\.  700  =  DCC,  7000=VII 


8=VIII.  80=LXXX.  800  =  DCCC".  8000  =  VI II 

33 


This  completes  the  method  by  addition.  Now  give  the  method 
by  subtraction: 

4  =  IV,  40  =  XL,  400  =  CD,  4000  =Tv 

9=IX,  90=XC,  900=CM,  9000=IX 

The  above  thirty-six  numbers  include  every  possible  combination 
below  10,000.  When  the  child  can  write  these  numbers  readily,  he 
should  have  no  difficulty  with  combinations  of  two  or  more  of  them. 

Example.  Write  3469  in  Roman  notation.  3469  =  3000  +  400  +  60 
+  9/  .3469  =  MMMCDLXIX.       1912  =  MCMXII. 

Ones  and  fives  have  characters  of  their  own.  Twos,  threes, 
sixes,  sevens,  and  eights  are  written  by  addition.  Fours  and  nines 
are  written  by  subtraction. 

Rule:  Beginning  at  the  left,  write  in  Roman  characters  the  num- 
ber represented  by  each  digit  in  turn.  That  is,  while  writing  the 
equivalent  to  one  digit,  do  not  pay  any  attention  to  the  other  digits. 

XI.  CANCELLATION. 

Teach  cancellation  progressively  as  follows.  See  illustrations 
below.  1.  The  same  numbers  in  dividend  and  divisor.  2.  All  num- 
bers except  1  cancelled.  3.  Numbers  in  dividend  divisible  by  numbers 
in  divisor.  First  strike  out  common  numbers  as  in  first  step.  4. 
Numbers  resulting  from  cancellation  cancelled.  The  order  here  (see 
below)  is  6  and  3  divided  by  3,  8  and  2  divided  by  2,  12  and  4  divided 
by  4.  5.     Numbers  in  dividend  and  divisor  divisible  by  the  same 

number.  The  order  is  6  and  3  divided  by  3,  8  and  2  divided  by  2,  12 
and  4  divided  by  4,  15  and  10  divided  by  5,  14  and  2  divided  by  2. 
6.  Result  in  fractional  form.     7.  Numerator  of  result  1. 

/•  z  z-  j* 

1  I  ivlvl  0X&X0    ,      ^A^.  ^ 


5  6  1 

2  1 

?    3     3    7  J  Z    Z  111 

^//-<.-XJ&XHL65  $X0XW_  4  JXW-  ± 

3A8X&    ~  3'XitXt£    J  0XfXJ&  4 

#  J2T  f    5  2.       Z 

When  possible,  in  the  solution  of  problems  time  should  be  saved 
by  cancellation.  State  all  successive  multiplications  and  divisions 
without  actually  performing  them,  then  solve  by  cancellation.  As  in 
other  work,  this  should  be  built  up  by  easy  steps. 

34 


1.  If  4  hats  cost  $12,  what  will  7  hats  cost?  At  first  have  prob- 
lems analyzed.  If  4  hats  cost  $12,  one  hat  will  cost  one-fourth  of  $12 
or  $2.  Have  this  placed  on  the  board  as  below,  emphasizing  that  the 
number  above  the  line  is  the  dividend  and  the  number  below,  the  di- 
visor. Continue  analysis.  If  1  hat  cost  $3,  7  hats  will  cost  7  times 
$3  or  $21.  This  will  appear  on  the  board  as  below  before  cancellation 
takes  place. 

2.  If  8  caps  cost  $4,  what  will  12  caps  cost?  Analize  and  cancel 
as  before. 

3.  If  12  bushels  of  apples  cost  $20,  what  will  9  bushels  cost? 

4.  What  will  7  turkeys  cost  if  I  pay  $12  for  8  turkeys? 

In  later  grades  much  more  difficult  problems  should  be  solved  by 
cancellation.  In  each  new  subject  admitting  the  use  of  cancellation, 
practice  should  be  given  in  the  statement  of  problems.  The  difficulty 
will  be  found  in  the  making  of  the  statement,  not  in  the  cancellation. 

5.  How  many  acres  in  a  field  80  rods  by  70  rods?     How  shall  we 
find  the  area  of  the  field?     70  times  80  sq.  rd.     Indicate  the  operation 
If  there  are  70  times  80  square  rods  in  the  field,  how  shall  we  find  the 
number  of  acres?     Divide  by  160  square  rods.     Indicate  the  division. 
Now  cancel. 

6.  What  will  914  pounds  of  wheat  cost  at  90  cents  per  bushel? 
What  is  the  missing  fact?  The  number  of  bushels.  How  shall  we 
find  it?  Indicate  this.  How  shall  we  find  the  cost  of  so  many  bush- 
els?    Indicate  this.     Cancel. 

$12  7x$12  12  X  $4  9x$20  7x$12 


4                4  8                   12 

70X80  sq.   rd.  914x90  cents. 

=  35.\35A.  

160  sq.  rd.  60 


35 


PART  THREE 

Denominate  Numbers  and  Practical 
Measurements 


I.   DENOMINATE  NUMBERS. 

Denominate  numbers  should  be  illustrated  by  exact  measure- 
ments. The  child  should  have  in  his  hands  the  foot  rule,  the  yard 
stick,  the  pint,  quart,  and  peck  measures,  the  pint,  quart,  and  gallon 
measures,  and  the  weights  to  be  used  with  the  balance  scales.  He 
should  discover  by  experiment  the  relation  between  the  different 
measures  and  build  up  his  own  tables  as  far  as  practicable.  He 
should  also  discover  the  difference  between  the  dry  and  the  liquid 
quart.     He  should  measure  and  weigh  various  objects. 

As  children  often  have  trouble  in  determining  whether  to  multi- 
ply or  divide  in  reduction  of  denominate  numbers,  the  following 
scheme  may  be  employed  for  a  time. 

Example:     Reduce  288  inches  to  feet. 

Write  the  equivalent,  then  write  the  number  of  inches  to  be  re- 
duced directly  below  the  12  in.  As  the  number  of  feet,  1,  is  less  than 
the  number  of  inches,  12,  the  number  of  feet  wanted  must  be  less  than 
the  number  of  inches,  288,  therefore  divide. 

12  in.  =  l  ft. 
288  in.  =  ?   ft. 
12  in.  =  l  ft. 
?    in.  =  288  ft. 

Example:     Reduce  288  ft.  to  inches. 

Here  evidently  the  number  of  inches  must  be  greater  than  the 
number  of  feet,  therefore  multiply. 

In  this  work  it  is  important  that  the  child  be  not  allowed  to  in- 
dicate that  288  ft.  multiplied  by  12  gives  3456  in.,  nor  that  288  ft. 
multiplied  by  12  in.,  nor  12  in.  multiplied  by  288  ft.  gives  3456  in.  He 
must  indicate  that  ]  2  in.  multiplied  by  the  abstract  number  288  gives 
the  desired  result.  288x12  in.  =  3456  in.  Ans.  The  first  example 
should  be  expressed  as  follows:     288  in.-:- 12  in.  =  24. ".24  ft.  Ans. 

Example:     Reduce  2  pk.,  5  qt.,  1  pt.  to  pints. 

If  this  example  is  worked  vertically,  do  not  allow  the  work  to 
appear  as  if  2  pecks  multiplied  by  8  equals  16  quarts,  nor  that  2  pints 
times  21  quarts  equals  42  pints.  Cancel  the  word  "qt."  after  the  21, 
thus  abstracting  the  number  and  consider  this  as  the  multiplier  and 
the  2  pints  as  the  multiplicand.  If  the  example  is  solved  horizontally, 
do  not  allow  the  following  false  statement:     2x8  qt.  =  16  qt. +  5  qt. 

36 


=  21  qt.;  for  2  times  8  qt.  equals  16  qt.  only,  not  16  qt.  plus  5  qt.  as 
the  statement  indicates.  Either  make  several  distinct  statements;  as, 
2x8  qt.  =  16  qt.  and  16  qt. +  5  qt.  =  21  qt.;  or  better  state  as  below: 

8qt.  2x8qt.  +  5qt.  =  21qt. 

2  21  X  2  pt.  +  1  pt.  =  43  pt.     Ans. 


16  qt. 
5qt. 


21  (qt.) 
2pt. 


Draw  a  line  through  this  "qt. 


42  pt. 
1  pt. 

43  pt.     Ans. 

In  connection  with  the  above  horizontal  statements,  if  it  has  not 
already  been  explained,  it  is  well  to  instruct  the  class  that  it  is  a  gen- 
erally accepted  mode  first  to  perform  all  indicated  multiplications  and 
divisions  in  the  order  in  which  they  occur,  then  to  perform  the  indi- 
cated additions  and  subtractions  in  order.  If  parentheses  are  used, 
operations  indicated  within  should  first  be  performed. 

See  Walsh,  pp.  129-140;  Bailey,  Lesson  23;  Brown  and  Coffman, 
Chap.  XII. 

II.  SQUARE  MEASURE. 

Example:     Find  the  area  of  a  surface  3  in.  by  5  in. 


3X5  sq.  in.  =  15  sq.  in.     Ans. 

As  there  are  5  square  inches  in  the  upper  row  and  there  are  3 
rows,  there  are  8  times  5  sq.  in.  in  the  whole  surface. 

37 


In  solving  such  problems,  have  the  class  make  the  illustrative 
diagrams  until  they  thoroughly  understand  the  process.  Under  no 
circumstances  allow  them  to  say  3  in.  times  5  in. 

Example:  If  a  rectangle  5  inches  long  contains  15  square  inches, 
how  wide  is  the  rectangle? 

15  sq.  inn- 5  sq.  in.  =  3. '.3  in.     Ans. 

If  there  are  5  sq.  in.  in  one  row  there  will  be  as  many  rows  as  5 
sq.  in.  is  contained  times  in  15  sq.  in. 

In  developing  the  number  of  square  inches  in  a  square  foot,  use 
the  above  method  and  have  each  child  make  a  diagram  of  a  square 
foot  or  144  square  inches  on  paper.  Thus  the  square  yard  also  should 
be  developed  and  placed  on  the  board. 

III.  CUBIC  MEASURE. 

Have  several  inch  cubes,  several  boxes  with  the  inside  dimen- 
sions in  exact  inches  if  possible,  and  one  box  exactly  one  cubic  foot 
inside  dimensions. 

Take  a  box,  say  3  in.  by  4  in.  by  5  in.     Place  n  inch  cubes  in  the 
box  in  a  row  along  one  edge.     Bring  out  that  there  would  be  4  rows 
of  5  cubic  inches  in  one  layer  of  blocks,  or  4  times  5  cu.  in.  or  20  cu 
in.;     that  in  three  such  layers  there  would  be  3  times  20  cu.  in.  or  60 
cu.  in. 

3x4x5  cu.  in.  =60  cu.  in.     Ans. 

Similarly  develop  the  number  of  cubic  inches  in  one  cubic  foot. 

IV.  PRACTICAL  MEASUREMENTS. 

In  practical  measurements  the  operations  should  when  possible, 
be  reduced  to  a  series  of  steps  to  be  taken  in  regular  order.  These 
steps  in  turn  may  often  be  reduced  to  a  set  form. 

V.  PERIMETER-FENCES. 

Bring  out  that  Perimeter  means  the  measure  around.  To  find 
the  perimeter  of  a  rectangular  figure  8  in.  by  12  in.,  add  the  length  of 
two  adjacent  sides:  12"  +  8"  and  multiply  by  2.  2(12"  +  8")  =40". 
the  Perimeter. 

Teach  at  once  that  a  number  before  or  after  a  parenthesis  indi- 
cates that  the  result  of  the  operations  within  the  parenthesis  should 
be  multiplied  by  the  number  or  numbers  without. 

In  dictating  examples  in  fencing,  etc.,  have  the  class  write  the 
following  blank  on  the  board  to  be  ready  to  take  down  the  numbers 
as  dictated:  2(  -J-).  As  the  example  is  dictated,  the  numbers  are 
placed  in  the  blank  spaces.  This  blank  will  be  found  useful  in  de- 
termining the  amount  of  fencing  required  to  fence  in  a  lot. 

VI.  AREA-ACRES. 

The  method  of  computing  areas  has  already  been  given  in  con- 
nection with  square  measure. 

38 


How  many  acres  in  a  farm  80  rods  by  140  rods? 
X  80  X 140 

=     .'.    A.  Ans. =  70. '.70  acres.     Ans. 

160  160 

Making  use  of  the  above  blank,  the  teacher  should  dictate  many 
examples  for  rapid  computation.  Insist  on  cencellation  where 
possible. 

VII.  PLASTERING. 

Take  a  chalk  box  about  8  inches  by  4  inches  by  4  inches.  Wha* 
is  the  perimeter  of  this  box?  Open  the  box  out  so  as  to  make  the 
four  sides  one  plane  surface.  What  is  the  length  of  the  four  sides? 
The  same  as  the  perimeter,  2(8"  +  4")  =24".  What  is  the  area  of 
the  four  sides?  2(8  +  4)4  square  inches.  Dictate  many  examples  for 
practice  in  finding  the  areas  of  four  sides  of  rooms.  As  the  area  of 
these  four  walls  has  been  found  in  square  feet,  how  shall  we  find  the 
number  of  square  yards  of  plastering? 

Bring  out  the  fact  that  the  openings  for  doors  and  windows  do 
not  have  to  be  plastered,  but  state  that  on  account  of  the  difficulty  of 
plastering  around  openings  plasterers  generally  deduct  only  half  of 
the  openings.  Only  such  deductions  should  be  made  as  are  directed 
in  each  example.  Develop  the  following  blanks  and  have  the  class 
place  them  on  paper  or  on  the  board  before  any  computations  are  be- 
gun. Insist  upon  the  placing  of  one  equality  sign  directly  under 
another. 

2(      +      )     —     =      .'.      sq.  yd. 

9 

X  =    .-. 


X     $  =     $         Ans. 

As  the  teacher  dictates  or  as  the  child  reads,  he  places  the  figures 
in  the  appropriate  blanks.  The  computations  aside  from  mental  pro- 
cesses and  cancellations  should  be  performed  below  the  line  and  the 
results  extended  properly. 

Example:  At  30  cents  per  square  yard  what  will  it  cost  to 
plaster  a  room  18'  x  22'  X  10',  allowing  one-half  of  100  sq.  ft.  for  the 
openings? 

2(18  +  22)10—50 

=83  13  number  of  sq.  yds. 


=44  number  of  sq.  yds. 


9 
18X22 

9 
Total  127  1  3  sq.  yds. 

127  1  3X$.30     =$38.20.     Ans. 

39 


In  the  second  statement  above,  cancellation  should  be  used  when 
possible.     Give  abundant  practice. 

VIII.   CEMENT  WALKS,  PAVING,  ETC. 

Use  the  second  statement  in  form  for  plastering.  When,  how- 
ever, a  walk  is  placed  on  two  or  more  sides  of  a  lot,  an  allowance  has 
to  be  made  for  one  or  more  corners;  one  if  on  two  sides,  two  if  on 
three  sides,  and  four  if  on  four  sides.  If  the  walk  is  on  the  lot,  the 
corners  are  deducted;  if  around  the  outside  of  the  lot,  the  corners  are 
added.     Illustrate  by  diagram. 

Example:  A.  Find  the  number  of  square  yards  in  a  4  foot  ce- 
ment walk  on  two  sides  of  a  lot  80  feet  by  40  feet.  B.  Around  two 
sides. 

A.  4(80  +  40 — 4)^-9  =  51  5/9  number  of  sq.  yd.     Ans. 

B.  4(80  +  40  +  4)  -^9  =  55   1/9  number  of  sq.  yd.     Ans. 

IX.  PAINTING. 

For  rectangular  rooms  use  the  same  form  as  in  plastering.  For 
irregular  buildings  find  the  total  area  in  square  feet  before  dividing 
by  9. 

X.  PAPERING. 

The  length  of  a  roll  of  wall  paper  is  24  feet.  Of  a  double  roll, 
48  feet.  The  width  is  generally  18  inches.  What  is  the  area  of  a 
single  foil?  Of  a  double  roll?  If  we  have  the  area  of  the  surface  to 
be  covered,  and  know  the  area  of  a  roll  of  paper,  how  shall  we  find 
the  number  of  rolls?  Is  it  possible  to  use  all  the  paper  of  a  roll? 
Some  paper  hangers,  allowing  for  waste,  figure  upon  three  single 
rolls  covering  100  square  feet  of  surface.  This  is  allowing  about  33 
square  feet  to  the  roll.  Therefore  divide  the  area  to  be  covered  by  33 
square  feet.  For  double  rolls  divide  by  65  or  66.  Salesmen  of  expe- 
rience assert  that  when  this  method  of  computation  is  used,  seldom 
if  ever  is  paper  returned  or  more  called  for.  The  strip  method  of 
computation  is  cumbersome  and  antiquated. 

Example:     At  75  cents  per  double  roll  what  will  it  cost  to  paper 
a  room  18' X  22' X  10',  allowing  100  square  feet  for  openings? 
•  » 
2(18  +  22)— 100 

=10+.  Ml  rolls. 

65 

18X22 


=  6+.'.   7     " 

65  

Total  18  rolls. 

18X$.75  =$13.50.     Ans. 


40 


If  an  unusual  width  of  paper  is  used,  a  different  divisor  must  be 
used. 

Before  beginning  an  example  in  papering,  the  following  blank 
should  be  placed  on  the  board  or  paper: 

2(    +    )    — 

=     .'.     rolls. 

33 

X 


rolls. 
X     $  =     $  Ans. 

For  double  rolls,  substitute  65  or  66  for  33.  Divisors  662A  and 
33^5  would  be  evon  better.  To  divide  by  66^,  divide  by  2/$  and  poinr. 
off  two  places.  To  divide  by  33^,  divide  by  1/3  and  point  off  two 
places. 

XI.  CARPETING. 

Though  in  many  homes,  for  sanitary  as  well  as  for  artisti'; 
reasons,  rugs  are  taking  the  place  of  carpets,  capets  are  still  gener- 
ally used. 

PREPARATION. 
If  this  room  is  18  feet  wide,  how  many  yards  wide  is  it?  How 
many  yard  sticks  placed  end  to  end  would  cross  the  room?  How 
many  measures  two  feet  long?  6  feet  long?  How  do  you  find  the 
number  of  measures  each  time?  By  dividing  the  width  of  the  room 
by  the  length  of  the  measure. 

STATEMENT  OF  AIM. 
Let  us  see  how  to  find  the  number  of  strips  of  carpet  required  to 
cover  a  floor. 

PRESENTATION. 
Place  a  strip  of  figured  wall  paper  on  the  floor.     If  this  paper 
were  2  feet  wide,  how  many  strips  shou'd  I  lay  side  by  side  to  cover 
the  floor?     If  it  were  3  feet  wide?     6  feet  wide? 

COMPARISON  AND  ABSTRACTION. 

In  each  case  how  did  you  find  the  number  of  strips  necessary? 
By  dividing  tae  width  of  the  room  by  the  width  of  the  strip. 

GENERALIZATION. 

Then  if  you  wished  to  carpet  any  room,  how  could  you  find  the 
number  of  strips  required?  To  find  the  number  of  strips,  divide  the 
width  of  the  room  by  the  width  of  a  strip.     Have  children  commit. 

41 


APPLICATION. 

If  the  room  is  6  yards  wide  and  the  strips  M  yard  wide,  how  shall 
we  find  the  number  of  strips?     Divide  6  yards  by  }i  yard. 
6--  34  =6  X  4/3  =  8. \8strips. 

If  a  room  is  17/3  yards  wide  and  the  carpet  %  of  a  yard  wide, 
find  the  number  of  strips. 

Would  a  clerk  cut  a  strip  lengthwise  for  you?  Then  how  many 
strips  would  you  be  required  to  buy?     Give  several  similar  examples. 

Note.  As  carpet  is  purchased  by  the  yard,  it  is  necessary  to  re- 
duce the  length  of  the  room  to  yards,  therefore  for  the  sake  of  uni- 
formity, reduce  all  dimensions  to  yards.  As  the  width  of  the  room 
is  generally  reducted  to  an  improper  fraction  before  division  by  ?4, 
leave  this  dimension  in  the  form  of  an  improper  fraction  of  a  yard. 

If  a  room  is  19  feet  wide,  what  must  we  do  with  the  19  feet  be- 
fore we  divide  by  Ya  yard?     Ans.  Reduce  19  feet  to  yards. 

19/3  yds. ^?4  yds.  =  19/3x4/3  =  76/9  =  8 +  .-.9  strips.  Give 
similar  examples. 

TO  FIND  THE  NUMBER  OF  YARDS. 

If  there  are  9  strips  each  8  yards  long,  how  many  yards  of  carpet 
will  be  required  to  cover  the  floor?     Etc. 

If  the  room  is  24  feet  long,  what  do  you  do  to  the  24  feet  before 
multiplying?     Ans.     Change  it  to  yards.     Give  examples. 

WASTE. 

To  illustrate  that  there  is  generally  waste  on  all  but  the  first 
strip,  use  the  figured  wall  paper. 

If  there  is  a  waste  of  1  foot  on  a  strip,  how  many  yards  waste? 
Ans.     1/3  yard.    8"  waste?     8/36  yd.  =  2/9  yd.  Etc. 

If  there  are  &  strips  and  a  waste  of  10  inches  on  a  strip,  how 
much  waste  in  all?     7x5/18  yds.  =  35/18  yds.  =  l  17/18  yd.     Etc. 

If  it  takes  56  yards  of  carpet  to  cover  the  floor  and  there  is  a 
waste  of  22/i  yds.  in  matching,  how  much  carpet  must  be  purchased? 
Etc. 

ORDER  OF  STEPS. 

I.  Place  on  diagram  all  dimensions  in  yards. 

II.  Find  the  number  of  strips  by  dividing  the  width  of  the  room 
by  the  width  of  one  strip. 

III.  Find  the  number  of  yards  required  to  cover  the  floor  by  mul- 
tiplying the  length  of  a  strip  by  the  number  of  strips. 

IV.  Find  the  number  of  yards  waste  by  multiplying  the  waste 
on  a  strip  by  the  number  of  strips  less  one. 

V.  Find  the  total  number  of  yards  that  must  be  purchased  by 
adding  the  last  two  items. 

VI.  Find  the  cost. 

Where  there  is  no  waste,  omit  steps  IV  and  V. 

42 


Before  dictation  or  work  on  paper  begins,   the  following  blank 
should  be  written: 
I. 

yd. 


yd, 


yd. 


waste 


yd. 


II. 

=          .'.  strips. 

III. 

X 

yd. 

yd. 

IV. 

X 

yd. 

yd. 

V. 

Total 

yd. 

VI. 

X$ 

=  $                   Ans 

Dictation  should  proceed  slowly  enough  to  allow  the  change  of 
all  dimensions  to  yards  as  the  numbers  are  given. 

Example:  At  $1.25  per  yard  what  will  be  the  cost  of  a  27  in. 
carpet  for  a  room  17  ft.  by  23  ft.,  allowing  8  in.  waste  for  matching 
a  strip? 

OPERATION. 


I. 


17  3  yd. 


23  3  yd. 


3-1  yd. 


waste  8  36  yd.  =2  9  yd. 


II. 

17      4     68 

3       3       9 

=    7  +  . '  .8  strip 

III. 

23 
8X—  yd. 

3 

=  61  1  3  yd. 

IV. 

2 

7X—  yd. 

9 

5 
=   1-  yd. 

V. 

Total 

62  8  9  yd. 

VI. 

8 
62- X $1.25 
9 

=$78.61.     Ans 
43 

XII.  BOARD  MEASURE. 

To  illustrate  one  board  foot  present  two  boards  each  6"  X  12"  X  1. 
Placing  these  side  by  side  we  have  a  board  l'Xl'X  1",  or  1  board 
foot.  Placing  them  end  to  end  we  have  a  board  2'x6"Xl"  still  1 
board  foot.  Placing  one  flat  upon  the  other  we  have  a  board  1'  X6"  X 
2"  again  1  board  foot.  Thus  develop  rule:  to  find  the  number  of 
board  feet  multiply  the  number  of  feet  long  by  the  number  of  feet 
wide  by  the  number  of  inches  thick,  or  multiply  the  length  in  feet  by 
the  width  in  inches  by  the  thickness  in  inches  and  divide  by  12  to 
reduce  the  width  to  feet.  To  find  the  number  of  thousand,  point  off 
three  places.     It  is  generally  best  not  to  cancel  the  thousand. 

XIII.  TO  FIND  THE  NUMBER  OF  BUSHELS  IN  A  BIN. 

Approximately   1  bushel  equals  5/4  cubic  feet,  or  1  cubic  foot 
equals  4/5  of  a  bushel;  hence  to  find  the  number  of  bushels  in  a  bin, 
multiply  4/5  of  a  bushel  by  the  number  of  cubic  feet  in  the  bin.     Ap- 
proximately how  many  bushels  in  a  bin  10  ft.  by  4  ft.  by  6  ft. 
10x4x6x4/5  bu.  =  192  bu.     Ans. 

Accurately  2150.42  cu.  in.  =  l  bu. ;  hence  to  find  the  number  of 
bushels  in  a  bin,  reduce  the  number  of  cubic  feet  to  cubic  inches  and 
divide  by  2150.42  cu.  in. 

(10x4x6x1728  cu.  in.) -s- 2150.42  cu.  in.  =  192.85/. 192.85  bu. 

XIV.  TO  FIND  THE  NUMBER  OF  GALLONS  IN  A  TANK 

OR  CISTERN. 

Accurately  231  cu.  in.  =  l  gallon.  To  aid  in  cancellation,  have 
children  commit  the  factors  of  231,  i.  e.  3,  7,  and  11.  In  all  of  these 
examples,  the  numbers  in  the  curves  should  be  placed  above  the  line 
and  the  divisor  below  the  line  and  cancellation  used  where  possible. 
How  many  gallons  in  a  tank  22  in.  by  15  in.  by  14  in.? 

(22 x  15X14)  -231  =  20/. 20  gal.     Ans. 

Approximately  1  cu.  ft.  =  7^  gallons.  How  many  gallons  of 
water  in  a  cistern  8  feet  in  diameter  if  the  water  is  7  feet  deep? 

22/7  X  4  x  4  x  7  X  15/2  gal.  =  2640  gal.     Ans. 


44 


PART   FOUR 
A.     Common  Fractions 


For  suggestions  regarding  teaching  fractions  see  Brown  and 
Ooffman,  How  to  teach  Arithmetic,  Chap.  XIII,  Bailey,  op.  cit.  Lesson 
21,  Walsh,  op.  cit.  Chap.  II,  McLellan  and  Dewey,  op.  cit.  Chap.  XIII. 

I.   PROPER  FRACTION  TAUGHT  OBJECTIVELY. 

A.  ]/>,  lA,  14,  etc.  B.  2/i,  }i,  etc.  See  especially  Brown  and 
Coffman,  pp.  183-4.  The  chief  thing  to  be  taught  in  class  is  the  writ- 
ing of  the  fraction  and  what  is  meant  by  the  numerator  and  the 
denominator,  the  denominator  indicating  into  how  many  parts  a 
number  has  been  divided,  the  numerator  indicating  how  many  of 
these  parts  have  been  taken.  Paper  cutting  will  be  found  as  good  a 
method  of  illustration  as  any.  The  terms  numerator  and  denominator 
should  at  once  be  defined  and  used.  Emphasize  that  the  denominator 
is  the  name  of  the  fraction,  that  is,  the  name  of  the  parts,  and  tells 
the  size  of  the  parts  or  into  how  many  parts  the  number  has  been 
divided.  Show  from  the  first  that  the  larger  the  denominator  the 
smaller  the  parts:  y2  is  larger  than  Yi,  which,  in  turn,  is  larger  than 
Va,  etc. 

II.   FINDING  A  FRACTIONAL  PART  OF  A  NUMBER. 

Y>  of  $12,  etc. 

This  step  is  given  in  connection  with  division  and  should  dissi- 
pate the  notion  that  a  fraction  merely  indicates  one  or  more  parts  of 
a  single  thing.  Thus  J4  of  a  number  may  indicate  $6  as  well  as  Yi 
of  a  dollar. 

HI.  THE  IMPROPER   FRACTION. 

Illustrate  by  piling  books  improperly,  large  upon  small  books. 

IV.  A   FRACTION  AN  INDICATED   DIVISION. 

Emphasis  should  be  laid  upon  the  fact  that  a  fraction  is  an  indi- 
cated division.  That  the  dividend  is  always  above  the  line  and  the 
divisor  below.  Have  children  state  which  term  of  a  given  fraction  is 
the  dividend  and  which  is  the  divisor.  Have  them  write  in  fractional 
form  12  divided  by  three,  etc.  Give  thorough  drill  till  firmly  fixed  in 
mind.  Here  also  the  principle  that  the  larper  the  denominator  the 
smaller  the  fraction  should  again  be  emphasized  and  illustrated  by 
such  fractions  as  12/3  =  4,  12/4  =  3,  etc. 

45 


V.  TO  EXPRESS  AN  INTEGER  AS  A  FRACTION. 

Call  on  children  to  tell  how  many  halves  in  1,  in  2,  etc.,  using 
ohjective  illustration  if  necessary. 

1  =  2/2  =  3/3  =  4/4,  etc.  2  =  4/2  =  6/3  =  8/4,  etc. 

VI.   REDUCTION  OF  A  FRACTION  TO  AN  EQUIVALENT 
FRACTION  WITH  A  LARGER  DENOMINATOR. 

Fold  two  equal  strips  of  paper,  one  into  halves,  the  other  into 
fourths.  Call  attention  to  the  fact  that  the  one  half  is  equal  to  two 
fourths  and  write  on  the  board.  1/2  =  2/4.  Similarly  show  that 
1/2  =  3/6,     1/3  =  2/6,    2/5  =  4/10. 

By  comparison  bring  out  that  in  each  case  both  numerator  and 
denominator  may  be  multiplied  by  the  same  number  without  chang- 
ing the  value  of  the  fraction.  Then  generalize  that  multiplying  both 
numerator  and  denominator  of  a  fraction  by  the  same  number  does 
not  change  the  value  of  the  fraction.  Give  thorough  drill  such  as: 
Multiply  both  terms  of  2/3  by  4.  What  fraction  results?  What  can 
you  say  of  the  relative  value  of  2/3  and  8/12?     Why?     Etc. 

SUB  LESSON— TO  REDUCE  A  FRACTION  TO  AN  EQUIVALENT 
FRACTION  WITH  A  GIVEN  DENOMINATOR. 
Reduce  3/5  to  an  equivalent  fraction  with  the  denominator  20. 
3/5=?/20.  By  what  do  you  multiply  5  to  obtain  20?  Then  if  you 
multiply  the  denominator  by  4,  what  must  you  do  to  the  numerator 
in  order  not  to  change  the  value  of  the  fraction.  Give  a  large  amount 
of  drill  in  such  reduction. 

VII.   REDUCTION  TO  LOWEST  TERMS. 

Principle  to  be  developed:  Dividing  both  numerator  and  denom- 
inator by  the  same  number  does  not  change  the  value  of  the  fraction. 
Either  present  objectively  as  in  the  last  case,  or  reverse  the  former 
process. 

S/4  =  6/8/.6/8  =  3/4;  2/3  =  6/9.\6/9  =  2/3;  1/5  =  2/10.'. 2/10  =  1/5. 
What  do  we  do  to  3/4  to  obtain  6/8?  Then  what  might  we  do 
to  the  6/8  in  order  to  reduce  back  to  3/4?  Do  we  change  the  value 
of  the  fraction?  Similarly  for  6/9  and  2/10.  Compare,  generalize, 
and  drill.  Also  give  drill  in  reduction  to  equivalent  fraction  with 
given  smaller  denominator;  as,  reduce  8/12  to  sixths.  Also  give  drill 
in  reduction  requiring  first  reduction  to  lower  then  to  higher  terms; 
as,  reduce  8/12  to  ninths.  8/12  =  2/3  =  6/9.  First  reduce  to  lowest 
terms;     reduce  result  to  required  fraction. 

VIII.  ADDITION  AND  SUBTRACTION  OF  SIMPLE  FRACTIONS 
WITH  A  COMMON  DENOMINATOR. 

Two  apples  plus  three  apples  =  how  many  apples?  Etc.     Two 
sixths  +  three  sixths  =  how  many  sixths?     2/6  +  3/6  =  5/6,  etc. 

46 


IX.  REDUCTION  OF  A  MIXED  NUMBER  TO  AN  IMPROPER 

FRACTION. 

Preparation — Reduction  of  a  whole  number  to  a  fraction,  and 
addition  of  fractions.  Let  us  see  how  to  reduce  3  2/5  to  fifths.  How- 
many  fifths  in  3?  How  many  fifths  in  15/5  and  2/5?  Then  to  how 
many  fifths  is  3  2/5  equal?  17/5.  Similarly  for  2  1/3  and  5  1/2. 
Compare,  generalize,  drill. 

X.  REDUCTION  OF  AN  IMPROPER   FRACTION  TO   A   WHOLE 

OR  TO  A  MIXED  NUMBER. 

Preparation — A  fraction  is  an  indicated  division,  and  division 
with  a  remainder. 

Presentation — Reduce  8/2  to  a  whole  number,  etc.  Reduce  9/2 
to  a  mixed  number.  9/2  =  4  with  a  remainder  of  1.  What  were  the 
nine?  Ans.  Halves.  Then  if  one  of  the  nine  halves  remains,  what 
is  it?     Ans.  1/2.     Then  9/2  =  what?     Ans.  4  1/2. 

XI.  REDUCTION  TO  A   COMMON  DENOMINATOR  WHEN  ONE 
DENOMINATOR  IS  A   MULTIPLE  OF  THE  OTHER   AND 

ADDITION  AND  SUBTRACTION  OF  SUCH  FRACTIONS. 

How  do  we  add  three  qts.  and  one  pt. ?  Change  1/2  to  fourths. 
Let  us  see  whether  we  can  add  3/4  and  1/2.  Why  not?  How  did  we 
add  3  qts.  and  1  pt. ?  To  what  did  you  say  1  2  is  equal?  Can  you 
add  it  to  3/4  now?  Similarly  add  2/3  and  1/6,  1/5  and  3-10,  etc. 
Compare,  generalize,  and  drill.  Have  examples  of  this  type  added 
mentally  as  soon  as  possible. 

XII.  REDUCTION  TO  LEAST  COMMON  DENOMINATOR  WHEN 
TWO  OR  MORE  DENOMINATORS  HAVE  A  COMMON  FACTOR 
AND  ADDITION  AND  SUBTRACTION  OF  SUCH  FRACTIONS. 

TWO  FRACTIONS. 

5/12  +  7/8.  See  illustration  below.  Divide  12  and  8  by  4,  th? 
largest  number  that  will  divide  both,  giving  3  and  2.  Multiply  the 
numerator  and  denominator  of  the  first  fraction  by  the  2.  giving  thr 
equivalent  fraction  10/24;  the  numerator  and  denominator  of  thr 
second  fraction  by  3,  giving  21/24.  In  this  and  in  previous  cases 
emphasize  the  idea  of  equivalent  fractions.  For  a  time  it  is  well  to 
have  children  draw  the  connecting  lines  as  guides  in  multiplication. 

THREE  OR  MORE  FRACTIONS. 
5/12  +  3/16  +  7/18.  Teach  this  in  the  old  way.  See  illustration 
below.  Note  that  the  divisor,  in  finding  the  least  common  denomina- 
tor, should  be  a  prime  number  unless  it  will  divide  all  thr  denomina- 
tors. For  example,  if  4  had  been  used  in  the  above  example  instead 
of  2  and  2,  the  factor  2  would  not  have  been  eliminated  from  18  and 

47 


the  common  denominator  288  would  have  been  found  instead  of  the 
least  common  denominator  144.  Multiply  2x2x3x4x3  =  144,  the 
new  denominator.     Divide  144  by  12  and  multiply  5  by  the  result,  etc. 

XIII.   REDUCTION  TO  A  COMMON  DENOMINATOR  WHEN  ALL 

DENOMINATORS  ARE  PRIME  TO  ONE  ANOTHER  AND 

ADDITION  AND  SUBTRACTION  OF  SUCH 

FRACTIONS. 

A.  TWO  FRACTIONS. 
2/3-j-i/5=?  See  illustration  below.  The  ordinary  method  of 
teaching  this  step  is  to  multiply  the  3  by  5,  then  to  divide  the  15  by  3 
to  find  5  which  is  already  before  us;  and  to  divide  15  by  5  to  obtain  3. 
This  is  waste  of  time.  Simply  multiply  2  and  3  by  5,  and  1  and  5  by 
3.  At  first  connect  the  numbers  to  be  multiplied  by  lines  and  have 
the  children  do  the  same,  but  this  should  soon  be  dropped. 


M-'r^rs    ukkkm+m+m 


3^T 


B.  THREE  OR  MORE  FRACTIONS. 
5/7 -|- 4/9 _|- 2/5.  Here  instead  of  dividing  the  common  denomi- 
nator, 315,  by  7,  9,  and  5  in  turn  to  get  45,  35,  and  63,  obtain  these 
numbers  by  multiplying  together  the  appropriate  denominators,  9  and 
5,  7  and  5,  7  and  9.  The  numerators  of  the  equivalent  fractions  may 
be  found  thus:     5(9x5)  =225,  4 (7x5)  =140,  and  2(7x9)  =126. 

XIV.  ADDITION  OF  MIXED  NUMBERS. 

A.  Sum  of  two  fractions— 1.     17^  +  1934+12^. 

Add  mentally  2/3  +  1/3  =  1,  1  plus  3/4  =  1  3/4.  Write  -4  and 
carry  1,  etc.  Drill  should  be  given  so  that  children  will  seek  the 
shortest  method. 

B.  Simple  fractions  that  can  be  added  mentally.     5^+7  +  8^4, 

etc. 

C.  More  difficult  fractions. 

See  illustrations  below.  Before  beginning  work,  write  the  word 
"Ans."  to  the  right  of  the  place  where  the  result  will  finally  appear. 
To  the  right  of  this  draw  a  vertical  line  to  keep  the  numerators  of 
the  equivalent  fractions  in  a  column.  In  the  last  example  note  that 
the  denominator  6  is  contained  in  the  denominator  12,  therefore  find 
the  least  common  denominator  of  12  and  9.  Divide  either  one  of  the 
two  by  3,  the  largest  number  that  will  divide  both,  and  multiply  the 
other  by  the  result,  9x(l2-^3)=36,  or  12x(9-^3)=36.     As  soon 

48 


as  the  least  common  denominator  is  found,  place  it  below  the  line  and 
to  the  right  of  the  word  "Ans.",  leaving  room  above  it  for  the  sum  of 
the  numerators.  Do  not  allow  the  pupils  to  place  the  mixed  num- 
ber 11  5/9  as  equal  to  20/36.  They  are  not  equal  and  it  is  a  waste  of 
time  to  write  the  36  more  than  once  before  placing  it  in  the  answer. 
So,  too,  the  numerators  are  more  easily  added  if  the  denominators  do 
not  intervene.  Reduce  5/9  to  36ths  and  write  the  numerator  20  to 
the  right  of  the  vertical  line  and  opposite  the  numerator  5,  and  have 
the  fraction  read  as  20/36.  Similarly  write  the  numerators  21  and  6, 
then  adding  the  three  numerators  place  the  result  above  the  common 
denominator  36.  The  children  may  then  write  the  equivalent  of 
47/36  in  its  simplest  form.  Place  the  fraction  11/36  in  the  space  re- 
served for  the  answer  directly  below  the  original  fractions  and  carry 
the  1  and  add  the  integers.  As  much  of  the  work  as  possible  should 
be  performed  mentally.  If  impossible  to  find  the  denominator  men- 
tally, place  the  denominators  at  one  side  and  find  the  least  common 
multiple;  however,  there  is  little  value  in  working  with  difficult  de- 
nominators. 

19  6  7 18  11  5  9 20 

12  2  3 14  16  7  12 21 

17  5  7 15  9  1/6 6 


5£  5/21  Ans.  47/21=2  5  21.  37  1 1  36   Ans.  47  36  =  1  11  36. 

XV.  SUBTRACTION  OF  MIXED  NUMBERS. 

The  following  seven  type  examples  are  arranged  progressively. 
When  possible  have  them  solved  mentally:  1.  Nothing  subtracted 
from  a  fraction.  2.  One  denominator  divisible  by  the  other.  3.  Two 
denominators  having  a  common  factor.  Divide  12  by  the  greatest 
common  divisor  4,  and  multiply  both  16  and  7  by  the  resulting  3. 
3x16  =  48  (least  common  denominator),  and  3x7  =  21.  The  other 
numerator  may  be  similarly  found:  5x  (16-5-4)  =20.  4.  No  common 
factor.  5,  6,  and  7  require  "borrowing";  or  add  1  to  both  minuend 
and  subtrahend.  In  7,  add  1  or  35/35  to  15/35  giving  50/35,  from 
which  we  subtract  28/35.  Also  add  1  to  401  making  it  402  and  then 
subtract. 

1.  2.  3.  4. 

24  3/4  17  2  3  8  7/16    21  48  2  3 

7'  5  5/9  5  5  12  20  17  12 

17  3/41  Ans.  1I[12  1  9lAns.       3  1  48  Ans.  1  48.  31   1  6  Ans. 

5.  6.  7. 

35 

44  44  411  3  7 15 

50 
3  5  17  3  5  4014  5 28 

43  2,5  Ans.       26  2  5  Ans.         9  22  35  Ans.  22  35. 

49 


XVI.  MULTIPLICATION  OF  A  FRACTION  BY  AN  INTEGER. 

A.  Multiplication  of  the  numerator. 

2x3  apples  =  6  apples,  etc.  2x3  sevenths  =  6  sevenths,  etc. 
2x3/7  =  6/7,  etc.  Principle  generalized:  Multiplying  the  numera- 
tor multiplies  the  fraction. 

B.  Division  of  the  denominator. 

2x3/8  =  6/8  =  3/4,  etc.  Teacher  erase  the  6/8.  What  can  you 
say  of  the  numerator  of  the  original  fraction  and  the  numerator  of 
the  resulting  fraction?  How  can  you  obtain  the  4  from  the  8  and  21 
Similar  questioning  for  other  fractions.  Principle  induced:  Dividing 
the  denominator  of  a  fraction  multiplies  the  fraction.  Induce  and 
have  children  commit  thoroughly  to  memory:  To  multiply  a  fraction 
divide  the  denominator  or  multiply  the  numerator.  Always  divide  if 
possible  to  save  the  time  of  reduction  to  lowest  terms.  Give  much 
drill  in  multiplication  of  simple  fractions  to  develop  quickness  in  de- 
cision as  to  whether  to  divide  or  to  multiply.  This  will  later  be  a 
great  saving  of  time. 

Sub  step. .  Multiplication  of  a  fraction  by  an  integer  that  is  iden- 
tical with  the  denominator  of  the  fraction.  4x1/4  =  1,  4x3/4  =  3, 
15x11/15  =  11,  etc.  Have  the  answer  written  at  once  without  writ- 
ing first  3/1  and  11/1.  Give  much  drill  till  the  children  can  give  re- 
sults without  the  slightest  hesitation. 

C.  Cancellation. 

Base  on  earlier  work  in  cancellation  of  whole  numbers.  12x5/16, 
etc.  Where  the  denominator  is  exactly  divisible  by  the  integer  or 
the  integer  is  exactly  divisible  by  the  denominator,  have  the  work 
performed  mentally  without  cancellation. 

XVII.  MULTIPLICATION  OF  AN  INTEGER  BY  A  FRACTION. 

Teach  by  analysis.  Find  2/3  of  $6.  Solve  orally  at  first.  1/3 
of  $6  is  $2;  2/3  of  $6  is  2  times  $2,  (1/3  of  $6),  or  $4.  Give  much 
oral  practice,  then  solve  on  the  board,  2/3  X  $6  =  $4,  accompanying 
the  solution  with  oral  analysis.  Rule  developed:  To  multiply  by  a 
fraction  divide  by  the  denominator  and  multiply  by  the  numerator. 
When  the  denominator  is  not  exactly  divisible,  cancel  if  possible.  The 
order  of  these  operations  should  be  determined  by  the  relation  of  the 
integer  to  the  terms  of  the  fraction.  In  3/4  of  8,  divide  first;  in  3/5 
of  8,  multiply  first.  Emphasize  that  when  2/3  of  6  is  found,  6  is 
multiplied  by  2/3.     This  understanding  is  essential  to  future  progress. 

XVIII.   MULEIPLICATION   OF  A   FRACTION   BY  A   FRACTION. 

Use  previous  cancellation  as  preparation. 

XIX.  MULTIPLICATION  OF  A  MIXED  NUMBER  BY  A 

MIXED  NUMBER. 

This  is  the  only  case  of  multiplication  where  reduction  to  im- 
proper fractions  is  preferable.     But  see  Bailey,  op.  cit.,  p.  101. 

50 


XX.   DIVISION  OF  A   FRACTION  BY  AN  INTEGER. 

Present  as  in  multiplication  of  a  fraction  by  an  integer.  15 
apples-^3  =  5  apples.  15/16-r-3  =  5/16.  A.  Principle:  To  divide  tho 
numerator  divides  the  fraction.  B.  Principle:  To  multiply  the  de- 
nominator divides  the  fraction.  Refer  to  the  principle:  The  larger 
the  denominator  the  smaller  the  fraction.  If  we  make  the  denomi- 
nator 4  times  as  large,  how  is  the  fraction  affected?  How  many 
times  smaller?  Then  multiplying  the  denominator  does  what  to  the 
fraction?  5/6 -f- 4  =  5/24.  Develop  and  have  children  commit  the 
rule:  To  divide  a  fraction  by  an  integer  divide  the  numerator  or 
multiply  the  denominator. 

Also  develop  the  principles:  A  change  in  the  numerator  pro- 
duces a  like  change  in  the  value  of  the  fraction,  a  change  in  the  de- 
nominator produces  an  opposite  change  in  the  value  of  the  fraction. 
Cive  much  practice  in  working  examples  requiring  judgment  as  to 
whether  it  is  bettei  to  divide  or  to  multiply  numerator  or  denomina- 
tor or  whether  to  cancel. 

XXI.  DIVISION  OF  AN  INTEGER  OR  OF  A  FRACTION 

BY  A  FRACTION. 

Here  it  is  probably  best,  though  not  essential,  to  teach  the  in- 
version of  the  divisor.  With  young  children  the  dictum  of  the 
teacher  will  doubtless  be  more  effective  than  a  philosophical  expla- 
nation. 

XXII.   DIVISION  OF  A  MIXED  NUMBER  BY  AN  INTEGER. 

A.  When  the  mixed  number  is  small,  reduce  to  an  improper 
fraction  and  divide. 

B.  When  the  mixed  number  is  large,  divide  directly  and  reduce 
the  remainder  to  an  improper  fraction  and  complete  the  division. 

2)1729^=861  2)1729^=864^.        The    remainder    1 

should  mentally  br  reduced  to  5/3  and  this  mentally  divided  by  2, 
which  gives  5/6.  4/3  divided  by  2  should  at  once  give  2/3,  not  4/6. 
In  such  examples  never  allow  reduction  to  an  improper  fraction. 

XXIII.   DIVISION  OF  A  WHOLE  NUMBER  BY  A  MIXED 

NUMBER. 

When  the  divisor  is  not  an  aliquot  part,  multiply  divisor  and 
dividend  by  the  denominator  of  the  fraction  and  divide  the  resulting 
numbers.  Basic  principle:  To  multiply  or  divide  both  dividend  and 
divisor  by  the  same  number  does  not  change  the  quotient.  This  prin 
ciple  should  be  developed  from  the  principle  that  multiplying  both 
numerator  and  denominator  of  a  fraction  by  the  same  number  does 
not  change  the  value  of  the  fraction;  and  from  the  fact  that  a  frac- 
tion is  an  indicated  division. 

51 


In  the  example  (1431-=- 175^),  what  can  we  do  to  both  dividend 
and  divisor  without  changing  the  quotient?  Then  multiply  both  by  3 
and  divide  results. 

At  first  it  may  be  well  to  place  in  fractional  form  till  the  relation 
is  clearly  seen.  Then  place  as  below.  At  first  allow  the  children  to 
put  down  the  multiplier,  but  as  soon  as  possible  have  them  remember 
that  the  multiplier  is  the  same  as  the  denominator  of  the  fraction. 

17^)1431     Multiply  both  by  3,  placing  result  two  lines  below. 

81  Ans. 
53     )4293 
424 
53 
53 

XXIV.  DIVISION  OF  A  MIXED  NUMBER  BY  A  MIXED  NUMBER 

Change  to  improper  fractions  and  divide. 

XXV.  MULTIPLICATION  OF  A  MIXED  NUMBER  BY  A 

FRACTION. 

Multiply  by  the  numerator  and  divide  by  the  denominator. 
4782  1/7  x  2/3.     See  below. 

XXVI.  MULTIPLICATION  OF  AN  INTEGER   BY  A   MIXED 

NUMBER. 

Do  not  allow  reduction  to  an  improper  fraction. 

9564  2/7  434  279 


4782  1/7 
2/3 

217 
17  2/3 

4783  3/7 
93 

3188  2/21  Ans. 

144  2  3 
1519 
217 

39  6/7 
14349 
43047 

3833  2/3  Ans. 

444858  6  7  Ans. 

XXVII.   DIVISION  OF  A  MIXED  NUMBER  BY  A  FRACTION. 

Time  will  generally  be  saved  by  inverting  the  divisor  and  multi- 
plying as  in  multiplying  a  mixed  number  by  a  fraction.  28  3/7-*- 2/3 
=  ?     Multiply  by  3/2  as  above. 

At  some  period  the  following  comparisons  should  be  made:  To 
multiply  a  fraction  by  an  integer,  multiply  the  numerator  OR  divide 
the  denominator.  To  multiply  by  a  fraction  multiply  by  the  numera- 
tor AND  divide  by  the  denominator. 

52 


B.     Decimal  Fractions 


I.  MEANING. 


1/10  =  . 1,       2/10  =  . 2,       9/10  =  .9,  etc. 
45/100  =  .45,        1/100  =  . 01,       7/100  =  .07,  etc. 
215/1000  =  . 215,        9/1000  =  .009,        17/1000  =  .017,  etc. 
There  are  as  many  places  to  the  right  of  the  decimal  point  as 
there  are  ciphers  in  the  denominator  of  the  common  fraction. 

II.   READING  AND  WRITING. 

At  first  as  the  teacher  dictates,  the  class  should  write  on  the 
board  in  the  common  fraction  form  and  then  in  the  decimal  form. 
The  ciphers  will  guide  them  in  placing  the  decimal  point.  Give  much 
practice  in  reading  and  writing  to  thousandths.  Beyond  thousandths 
"numeration"  may  be  necessary.  0.001,786 — tenths,  hun- 
dredths, thousandths,  ten-thousandths,  hundred-thousandths,  million- 
ths — 1,786  millionths.  See  that  the  -ths  is  annexed  in  all  cases. 
Insist  on  correct  wording  when  reading  beyond  thousandths.  Do  not 
allow  the  use  of  the  word  "of"  in  reading  .4,612.  Read  exactly  as 
the  common  fraction  4,612/10000  is  read.  Emphasize  the  use  of  the 
word  "and"  to  indicate  location  of  the  decimal  point,  and  the  use  of 
the  decimal  point  to  locate  units.  It  is  well  to  teach  both  of  the  fol- 
lowing forms:  0.721  and  .721,  as  both  are  sooner  or  later  met  with. 
First  teach  reading,  then  writing  of  many  such  numbers  as  the  fol- 
lowing: 1,000,000.000,001;  78,000,004.06,078;  317,009,000.7,048; 
2,002,002.2,000,002.  After  multiplication  and  division  of  decimals  by 
moving  the  decimal  point  has  been  taught,  dictate  numbers  to  be 
written,  have  the  class  divide  or  multiply  by  10;  100;  or  1,000,  and 
write  the  result  in  words.  See  that  such  words  as  twenty-five  and  all 
the  parts  of  the  denominator  are  separated  by  a  hyphen;  as,  seventy- 
eight  million,  four,  and  six  thousand,  seventy-eight  hundred- 
thousandths. 

III.  PLACE  VALUE. 

Analyze     numbers     as     follows.  2987.456  =  2000  +  900  +  80 

7 +  .4 +  .5 +  .006.  Show  that  in  33.33  each  3  is  ten  times  as  great  in 
value  as  the  3  to  its  right,  and  one-tenth  as  great  in  value  as  the  3  to 
its  left.        3  tens  or  30  =  10x3.  3  =  1/10  of     30  or     of     3     tens 

3-v- 10  =  3/10  or  .3.  .3  or  3/10  =  10  X  3/100  (or  .03).  .03  =  1/10  of 
.3.     Etc. 

IV.  CIPHER  AT  THE  RIGHT  OF  A  DECIMAL. 

Show  that  .20  may  be  read  20  hundredths  and  that  it  is  equal  to 
.2.      20/100  =  2/10. '..20  =  .2.         Similarly      .2000  =  .200  =  .20  =  .2,   etc. 

53 


Generalization:  Annexing  ciphers  to  the  right  of  a  decimal  does  not 
change  the  value  of  the  decimal.  Similarly  show  that  1.2  may  be 
read  one  and  two  tenths  or  12  tenths.  1.2=1  2/10  =  12/10.  Similarly 
12.5  =  12  5/10  =  125/10  =  125  tenths.      4.73  =  4  73/100  =  473/100,  etc. 

V.  REDUCTION  OF  A  FRACTION  TO  A  DECIMAL. 

Preparation.  A.  A  fraction  is  an  indicated  division.  B.  Division 
of  United  States  money.  C.  3/4  =  75/100  =  .75.  Presentation.  Re- 
duce 3/4  to  a  decimal.  As  3/4  means  to  divide  3  by  4,  let  us  perform 
the  indicated  division  as  we  divide  in  United  States  money. 

$.75  .75 


4)$3.00  4)3.00 

Therefore  34  =.75,  the  same  result  we  found  in  c.  Take  similar 
examples  and  draw  the  Generalization:  To  reduce  a  fraction  to  a 
decimal,  divide  the  numerator  by  the  denominator  and  point  off  as  in 
United  States  money. 

VI.  REDUCTION  OF  A  DECIMAL  TO  A  FRACTION. 

Write  the  decimal  as  a  common  fraction  and  reduce  to  lowest 
terms.  In  the  second  example,  multiply  the  numerator  and  denomi- 
nator of  the  complex  fraction  by  3  and  reduce  the  result  to  lowest 
terms. 

.75  =  75/100  =  3/4.        .13*3    =  13^/100  =  40/300  =  2/15. 

VII.  ADDITION   AND    SUBTRACTION    OF    DECIMALS. 

Principle:  Only  like  numbers  can  be  added,  units  only  can  be 
added  to  units,  tenths  only  to  tenths,  etc.,  therefore:  RULE:  Place 
one  decimal  point  under  another,  keeping  units  under  units,  tenths 
under  tenths,  etc.  Example,  from  17  subtract  .007.  Emphasize  that 
a  point  is  understood  to  the  right  of  units. 

VIII.  MULTIPLICATION   OF   DECIMALS. 

Multiplication  of  a  decimal  by  an  integer  is  easily  developed  by 
referring  to  multiplication  of  United  States  money.  Develop  multi- 
plication of  a  decimal  by  a  decimal  as  follows:  .5  X  .7  =  5/10  X  7/10  = 
35/100  =  . 35.  .   3  X. 08  =  3/10x8/100  =  24/1000  =  .024.         .45X.79 

=  45/100  =  79/100  =  3555/10000  =  .3555.       Etc.       Compare,     abstract, 
and  generalize. 

IX.  DIVISION  OF  DECIMALS. 

A.   Decimal  by  an  integer. 

Base  on  division  of  United  States  money.  Place  the  decimal 
point  in  the  result  directly  above  the  point  in  the  dividend. 

54 


2.39  .12  1.2  .012 

2)4.78  144)17.28  144)172.8  144)1.728 

It  is  well  to  place  the  point  in  the  result  before  the  division  be 
gins.        The  first   significant   figure  in   the   result   should  be  directly 
above  the  right  hand  figure  in  the  first  subtrahend.     All  the  figures 
should  be  in  strictly  vertical  columns.     The  teacher  should  insist  upon 
exactness  in  this  respect,  as  it  is  essential  to  accuracy. 
B.   Division  by  a  decimal. 

Use  the  Austrian  method.  Principle:  Division  of  both  dividend 
and  divisor  by  the  same  number  does  not  change  the  quotient.  See 
lesson  on  fractions,  Section  XXIII,  for  the  development  of  this  prin- 
ciple. Method:  Move  the  decimal  point  in  the  divisor  far  enough  to 
the  right  to  make  the  divisor  a  whole  number,  move  the  point  in  the 
dividend  the  same  number  of  places  to  the  right,  perform  the  division 
with  the  resulting  numbers.  1.728 -=-1.44  =  172.8-=- 144.  172.8-=-. MI 
=  ?  First  move  point  in  the  divisor  three  places  to  the  right,  then 
move  the  point  in  the  dividend  three  places  to  the  right  annexing 
sufficient  ciphers,  place  the  point  in  the  result  directly  above  the  new 
point  in  the  dividend,  divide  placing  the  first  figure  1  in  the  result 
directly  above  the  figure  2  in  the  dividend  as  the  right  hand  4  of  the 
first  subtrahend  144  is  also  placed  below  the  2.  In  dividing  by  .40 
move  the  point  only  one  place  to  the  right  and  divide  by  short  division. 
Divide  $120  by  $.40.  In  dividing  by  such  numbers  as  20,  300,  4,000. 
etc.,  move  the  decimal  point  to  the  left  of  the  ciphers  in  the  divisor, 
the  same  number  of  places  to  the  left  in  the  dividend,  and  divide  by 
short  division.     476-:- 400. 

1.2  1   200.  30  0.  1.19 


1.44)1.728  .144)172.800-  .4'0)120.0'  4*00)476 

X.  FRACTION  AT  THE  END  OF  A  DECIMAL. 

.12K>  =.125;     A%=Aiy4  =A12V2  =.4125; 
^=.3^=.33H=.333^,  etc. 
^=.6^=.6G^,  etc.      .16*$  =.166-.?,  etc. 

Give  sufficient  practice  to  familiarize  pupils  with  these  equiva- 
lents, that  they  may  be  able  to  read  and  write  them  accurately. 

XL  APPLICATION  OF  DECIMALS  IN    DIVISION   BY  A 

MIXED  NUMBER. 

Divide  7828  by  13 y3.  Multiply  both  numerator  and  denominator 
by  3,  move  the  decimal  points  in  the  results  one  place  to  the  left,  and 
divide  by  short  division. 


4.0)2348.4 


587.1  Ans. 
55 


XII.  ALIQUOT  PARTS. 

Pupils  should  become  so  familiar  with  the  following  equivalents 
that  one  immediately  suggests  the  other.  They  should  be  encouraged 
to  use  the  equivalent  most  economical. 

16ths     12ths  llths  lOths  9ths  8ths    7ths    6ths    5ths    4ths    3ds    half 
1/16  .06  1/4 


1/12 

.08  1/3 

1  11 

.09  1/11 

1/10 

.10 

1/9 

.11  1/9 

2/16 

1/8 

.12  1/2 

] 

;7 

.14  2/7 

212 

1/6 

.16  2/3 

2  10 

1/5 

.20 

4  16 

3/12 

2/8 

1/4 

.25 

3/10 

.30 

4/12 

39 

26 

1/3 

.33  1/3 

6/16 

3/8 

.37  1/2 

4/10 

2/5 

.40 

8/16 

6/12 

5/10 

4/8 

3/6 

2,4 

1/2 

.50 

6/10 

3/5 

.60 

10/16 

5/8 

.62  1/2 

8/12 

6/9 

4/6 

2/3 

.66  2/3 

7/10 

.70 

12/16 

9/12 

6/8 

3/4 

.75 

8/10 

45 

.80 

1012 

5/6 

.83  1/3 

14/16 

7/8 

.87  1/2 

9/10 

.90 

XIII.  MULTIPLICATION  BY  ALIQUOT  PARTS. 

Considerable  emphasis  should  be  laid  on  this  process: 

A.   By  aliquot  parts  of  1.     Multiply  by  the  equivalent  fraction. 
.33^X465=(^X465);  .14  2/7  X  212=  (1/7  X  212) ;  .66^ 

X  516=^x516;     .12^  X144=(^  X144). 

56 


B.  By  aliquot  parts  of  10,  100,  or  1000.  Multiply  by  the  appro- 
priate fraction  and  also  by  10,  100,  or  1000  as  the  case  may  be;  that 
is,  multiply  by  the  appropriate  fraction  and  annex  one,  two,  or  three 
ciphers.  12^X144=^x100x144  =  1800.  66^x516=^X5160 
=  34400.  1.25X  144=  (%  X  10x144)  =180.  6.6^X516=^X5160 
=  3440.  125  X  144=  (%  X  1000x144)  =1800.  666^x516  = 

5<3  x  516,000  =  344,000.  In  performing  the  operations,  do  not  write 
down  the  numbers  within  the  curves  above,  but  compute  mentally. 
Give  much  practice. 

XIV.  DIVISION  BY  ALIQUOT  PARTS. 

A.  By  aliquot  parts  of  1.  Divide  by  the  equivalent  fraction. 
252-.33^  =  (252-J4)=756.  144-12J^  =  (144-  %  )  =1152.  516 
-^.66^=516x3/2  =  774. 

B.  By  aliquot  parts  of  10,  100,  or  1000.  Divide  by  the  appropri- 
ate fraction  and  by  10,  100,  or  1000;  that  is,  divide  by  the  appropriate 
fraction  and  point  off  one,  two,  or  three  places. 

1779-1634  =  (1779-^6  ^100)  =106.74. 

279+- 333^  =  (279-  V3  -1000)  =.837. 

7421  —  3^  =2226.3       144  — 12^4  =  (144-  ys  -100)  =  11.52. 

516-66^  =5.16x3/2  =  7.74.         144-1.25  =  11.52. 

516-6.66^=51.6x3/2  =  77.4.       144-125  =  1.152. 

516-666^  =  .516x3/2  =  . 774. 

C.     Fractional  Relations 


I.  TO  FIND  A  FRACTIONAL  PART  OF  A  NUMBER. 

Find  Vi  of  $18.     Find  .7  of  $250.     Find  .17  of  $500. 
Fraction   x  whole  =  part. 
Factor  X  factor  =  product. 
Y3X%  18  =  $12. 
.7X  $250  =  $175. 
.17  X  $500  =  $85.00. 

II.  TO  FIND  WHAT  PART  ONE  NUMBER  IS  OF  ANOTHER. 

Preparation: 

Factor     Factor     Product 
2x3      =      6 
Given  the  product  6  and  one  factor  3,  how  do  we  find  the  other 
factor?     In  general,  how  do  we  find  the  other  factor  when  one  factor 
and  the  produce  are  given? 

57 


Ans.  Divide  the  product  by  the  given  factor  to  find  the  other 
factor. 

Factor     Factor     Product 

y3      X      $18   =   $12. 
.7  X  $250  =  $175. 
.17X$500  =  $85.00. 
Statement  of  aim:     Let  us  see  how  to  find  what  part  one  number 
is  of  another.     $12  is  what  part  of  $18?     $175  is  what  part  of  $250? 
$85  is  what  part  of  $500? 
Presentation: 

Statement  of  relation: 

Factor     Factor     Product 
?       x      $18      =      $12. 
?       X    $250      =      $175. 
?       X    $500      =      $85. 
Given  the  product  $12  and  the  factor  $18,  how  shall  we  find  the 
other  factor?     The  product  $175  and  the  factor  $250?     Etc. 

Ans.  Divide  the  product  $12  by  the  factor  $18.  Divide  the  pro- 
duct $175  by  the  factor  $250.     Etc. 

Statement  for  solution:  12/18  =  2/3  Ans.  $175-^$250  =  .7 
Ans.     $85 -h  $500  =  .17  Ans. 

In  finding  what  part  one  number  is  of  another  it  is  some  times 
preferable  to  place  the  division  in  fractional  form  and  reduce  to  low- 
est terms  as  above. 

The  two  satements,  the  statement  of  relation  and  the  statement 
for  solution  may  be  used  whenever  these  examples  are  to  be  worked; 
or  the  teacher  may  develop  the  rule  to  divide  the  part  by  the  whole 
to  find  the  fraction. 

III.  TO  FIND  THE  WHOLE    WHEN    A    PART    AND    ITS  FRAC- 
TIONAL RELATION  TO  THE  WHOLE  ARE  GIVEN. 

Preparation:  The  product  divided  by  the  factor  gives  the  other 
factor. 

Factor     Factor     Product 
y3     x    $18    =   $12. 
.17    X    $500   =    $85. 
.7    X    $250    =    $175. 
Statement  of  aim:     Let  us  see  how  to  find  the  whole  when  the 
part  and  its  relation  to  the  whole  are  given.     $12  is  2/z  of  how  many 
dollars? 

Presentation. 

Statement  of  relation: 

Factor     Factor     Product 
Vi     X      $   ?     =   $12. 
.7     X      $?    =    $175. 
.17     x      $?    =    $85. 

58 


Statement  for  solution: 

$12     -5-      Vi    =   %  Ans. 

$175   ■+■    .7  =  $  Ans. 

$85      ■+•      .17  =  $  Ans. 

As  in  the  preceeding  case,  the  two  statements  may  be  used  or 
the  teacher  may  develop  the  rule  to  divide  the  part  by  the  fraction  to 
find  the  whole.  Call  attention  to  the  fact  that  when  "of"  is  used  to 
indicate  multiplication,  the  whole  always  follows  the  word  "of". 
Give  an  abundance  of  drill  on  these  three  cases  of  fractional  relations 
as  a  preparation  for  Percentage.  Have  the  children  thoroughly  un- 
derstand that  when  the  part  is  wanted  they  always  multiply,  hence 
the  part  is  the  product.  When  the  part  is  given  it  is  always  divided 
by  whichever  factor  is  given  to  find  the  other  factor.  At  first  the 
statements  of  relation  and  solution  should  both  be  written  before  the 
example  is  worked,  a  blank  space  being  left  for  the  insertion  of  the 
answer. 

.7  X$?  =$17.50.  ?X$18  =  $12. 

$175--. 7  =  $  Ans.        $12/$18  =  2/3.        Ans. 


D.     Oral  Analysis 


The  old  method  of  oral  analysis  should  be  used  in  the  solution 
of  simple  problems,  though  it  should  not  be  allowed  to  crowd  out  the 
work  in  fractional  relations  which  is  a  preparation  for  Percentage. 

Example:  Find  Y,  of  $1200.  Solution:  \\  of  $1200  is  $300, 
and  Yi  of  $1200  is  3  times  $300  or  $900. 

Example:  $900  is  Ya  of  how  many  dollars?  Solution:  %  of 
the  money  (which  is  lA  of  Y  of  the  money),  is  lA  of  $900  or  $300, 
and  4/4  of  the  money  is  4  times  $300  or  $1200. 

As  oral  work  should  be  given  daily,  a  class  should  soon  become 
proficient  in  analysis.  The  statement  within  the  curves  is  explana- 
tory and  should  not  be  required  in  the  solution,  though  the  children 
should  be  able  to  give  it  if  requested. 


59 


PART  FIVE 

Percentage 


I.  PREPARATION  FOR  PERCENTAGE. 

As  a  preparation  for  Percentage,  aliquot  parts  and  fractional 
relations  should  be  thoroughly  mastered.  Much  use  should  be  made 
of  fractions  and  decimals  with  denominators  of  100  and  of  the  equiv- 
alent fraction  in  its  lowest  terms.  While  decimals  are  a  special  case 
in  fractions  with  denominators  of  ten  or  powers  of  ten,  Percentage  ^ 
is  a  special  case  in  decimals  with  denominator  100.  This  should  be 
made  clear  to  the  class.  The  following  questions  will  illustrate  the 
method  of  procedure: 

12  is  what  part  of  16?  How  many  hundredths?  write  the  frac- 
tion as  a  decimal.  What  part  of  15  is  10?  How  many  hundredths? 
What  part  of  7  is  6?  Change  to  the  decimal  form  expressing  it  as 
hundredths.     Etc. 

Find  .75  of  $84.  Find  y4  of  $84.  Compare  the  results.  Find 
.25  of  $738.  Find  %  of  $738.  Compare  results.  Find  .66%  of  $18. 
To  what  fraction  is  .66%  equal?  Find  %  of  $18.  Compare  results. 
Which  is  the  shorter  way?  Find  .16%  of  $72.  By  what  simple  frac- 
tion may  we  multiply  and  obtain  the  same  result?  Find  .07  of  $17. 
Is  there  any  simple  fractional  equivalent  of  .07?  Then  what  shall  we 
multiply  by  in  this  case?     Etc. 

$15  is  .05  of  how  many  dollars?  $18  is  M2A  of  how  many  dol- 
lars? What  simple  fraction  may  we  substitute  for  the  decimal?  Do 
it  and  solve.     Etc. 

Drill  until  the  children  can  readily  solve  simple  problems  in 
decimals  and  fractions  with  denominators  of  100,  using  the  decimal 
when  there  is  no  simple  fractional  equivalent,  and  exchanging  frac- 
tions and  decimals  without  hesitation  when  it  is  advantageous  to  do 
so.  When  the  class  understand  the  precess,  tell  them  that  they  have 
been  working  examples  in  "Percentage",  a  name  applied  to  examples 
in  fractions  and  decimals  in  which  the  denominator  is  100. 


II.  INTRODUCTION  OF  THE  TERMS  PER  CENT,  RATE  PER 

CENT,  AND  RATE. 

As  soon  as  the  children  are  familiar  with  the  method  of  working 
examples  in  hundredths,  the  terms  per  cent,  rate  per  cent,  and  rate 
should  be  introduced.  The  class  should  become  familiar  with  these 
terms  before  the  introduction  of  the  term  percentage  that  the  two 
terms  may  not  be  confused  as  sometimes  happens  when  they  are  in- 
troduced together.     Tell  the  children  that  as  the  term  per  cent  is  used 

60 


it  means  exactly  the  same  as  the  word  hundredths.  Also  teach  the 
symbol  for  per  cent  and  have  the  class  write  the  various  equivah 

75  percent  =  7595  =s=. 75=75/100=  H. 

62^  percent  =  62Y  '/  =  .62^  =.625=  $|. 

7  per  cent  =7';  =.07  =  7/100.     Etc 

Emphasize  that  as  per  cent  means  hundredths,  when  a  per  cent 
is  written  as  a  decimal,  two  decimal  places  are  necessary,  (live  much 
practice  in  writing  the  per  cent  as  a  decimal  of  two  places,  then  have 
the  fractional  part  of  the  per  cent  also  written  as  a  decimal.  Read 
Vi  '.',  ,    '  j  ' < ,  etc.,  as  l/i  of  one  per  cent,   '  \  of  one  per  cent,  etc. 

62^  </<  =.6254  =.625.        7'  ,  ';  --.07' ',  =.0725. 

y2  %  =  .oo  y2  =  .005.      y  y,  =  .00  %  =  .0025. 

Vi  <A  =  .00  Vi  =  .0033 1/3  =  .003333  Ys ,  etc. 

Children  should  also  acquire  facility  in  changing  such  decimals 
to  the  per  cent  form.  To  demonstrate  to  the  class  the  decimal  equiv- 
alent of  the  fractional  part  of  a  per  cent,  use  the  following  subtrac- 
tion: 

(a)  11  y, '■  =.ny2  =.115. 

(b)  179?  =.17=.17. 

(c)  \i  <:,  =.00^  =.005.     Subtracting  b  from  a. 

III.  INTRODUCTION  OF  THE  TERMS  PERCENTAGE  AND  BASE 

Review  the  words  whole  and  part  as  used  in  fractional  relations 
and  tell  the  class  that  in  the  study  of  Percentage  new  terms  are  used 
for  the  words  part  and  whole.  For  the  whole  we  use  the  term  Base, 
and  for  the  part,  the  term  Percentage.     Write  on  the  board: 

$8  is  25 %  of  $32.  $8  is  %  of  $32.  $8  is  25^  of  what  quantity? 
Find  25  </<  of  $32.     $8  is  what  per  cent  of  $32? 

Ask  such  questions  as:  What  is  the  rate?  What  is  the  whole? 
Then  what  shall  we  call  this  in  percentage?  What  is  the  part? 
What  is  the  new  word  that  we  have  learned  for  part?  Or  in  other 
examples:  Can  you  tell  which  is  the  percentage?  If  not,  ask  your- 
self which  is  the  part?  Can  you  tell  now?  Which  is  the  base?  Or: 
What  do  we  call  the  25';  ?  What  is  the  $32?  Is  it  the  part  or  the 
whole?  Then  what  shall  we  call  it?  What  is  called  for?  Etc.  Con- 
tinue such  questioning  till  the  children  can  tell  at  once  which  term  to 
apply  to  each  number  without  first  asking  themselves  which  is  the 
'■art,  etc.  It  is  not  necessary  to  have  these  examples  worked  as  the 
drill  is  in  the  application  of  terms.  It  is  important  that  the  class  un- 
derstand the  terms  and  can  apply  them  before  they  try  to  so 
problems. 

IV.  GIVEN  THE  BASE  AND  RATE  TO  FIND  THE  PERCENT  \(! 

Preparation: 
Find  .17  of  $59.     Find  .73  of  $714.     Find  .08  of  $216.     What  new 

61 


way  have  we  learned  to  express  .17?  Ans.  17 ',',  Similar  questions 
for  .73  and  .08.  What  do  you  call  the  17%,  73%,  and  8%?  The 
Rate  per  cent.    What  do  we  call  the  $59,  $714,  and  $216?  The  Base. 

Sub  step — Statement  of  aim. 

Let  us  see  now  how  to  find  the  Percentage  when  the  Rate  and 
Base  are  given. 

Presentation: 
Let  us  find  17%  of  $59.  What  did  you  say  the  $59  is?  The 
17%  ?  What  did  we  find  was  the  equivalent  of  179c?  And  how  did 
you  find  .17  of  $59?  Then  how  shall  we  find  17%  of  $59?  Ans.  Mul- 
tiply .17  times  $59.  Do  it.  What  do  we  call  the  result?  Ans.  The 
Percentage.  How  did  we  get  it?  Ans.  We  multiplied  .17  times  $59. 
But  what  is  the  17%  that  you  multiplied  by?  The  $59?  Then  how 
did  you  get  the  Percentage?  Ans.  We  multiplied  the  Rate  times  the 
Base.     Similarly  work  and  question  regarding  two  other  examples. 

Comparison. 
How  did  you  find  the  Percentage  in  each  of  the  three  examples 
you  worked?     We  multiplied  the  Rate  times  the  Base. 
Generalization: 
Who  will  give  me  a  rule  for  working  all  examples  when  the  Rate 
and  Base  are  given  and  we  wish  to  find  the  Percentage?      Rule:     To 
find  the  Percentage  when  the  Rate  and  Base  are  given,  multiply  th? 
Rate  times  the  Base.     If  we  let  R  stand  for  Rate,  B  for  Base,  and  1 
for  Percentage,  who  will  write  this  rule  as  a  formula. 
RxB  =  P. 

This  we  shall  call  the  Formula  for  Percentage  and  shall  read  it: 
The  Rate  times  the  Base  equals  the  Percentage. 

Application: 
Send  class  to  board  to  write  formula  and  interpret  it.  Each 
member  of  the  class  should  leave  the  formula  written  at  the  top  of 
the  board  and  as  the  teacher  dictates  examples  to  be  worked,  the 
numbers  should  be  written  below  the  appropriate  letter  in  comp'ete 
statement  form.  Example:  The  Base  is  $218,  the  Rate  15%,  find 
the  Percentage.  Each  number  is  properly  placed  as  dictation  pro- 
ceeds and  the  following  should  appear  on  the  board  before  work 
begins: 

R     X      B     =     P. 
.15     X    $218    =      $  Ans. 

Insist  on  complete  statement  before  work  begins.  The  example 
should  then  be  worked  below  and  the  result  placed  in  the  blank  left 
in  the  statement. 

Assignment: 
Problems  should  be  read  from  the  text  book,  the  numbers  iden- 
tified by  the  class  in  terms  of  percentage,  and  the  method  of  solution 
stated.     The  probliems  should  then  be  worked  during  the  study  period 

62 


V.  GIVEN  THE  BASE  AND  THE  PERCENTAGE  TO   FIND 

THE  RATE. 

Review  fractional  relations  to  find  what  part  one  number  is  of 
another.  Place  formula  on  the  board  and  below  its  appropriate  letter 
each  number  of  the  example.  $12  is  what  per  cent  of  $36?  The  class 
should  write  the  Formula,  RxB  =  P,  and  below  it  the  Statement  of 
relation,  ?  x$36  =  $12.  As  the  Percentage  is  the  result  of  multiply- 
ing the  Rate  and  the  Base,  what  is  the  Percentage  of  the  Rate  and 
Base?  Ans.  The  product.  Then  given  the  product  $12  and  the  factor 
$36,  how  shall  we  find  the  other  factor,  the  Rate?  State  it.  The  ex- 
ample should  now  appear  on  the  board  as  follows: 

(Formula)  R    X        B      =      P 

(Statement  of  relation)         ?     X     $36     =     $12 
(Statement  for  solution)  $12  $36  =1  '3=33  1  3>'  Ans. 

Example:     $28.63  is  what  per  cent  of  $409? 

R       X       B       =        P 

R       X     $409    =    $28.63 
$28.63-*-  $409    =       %       Ans. 

Solve  and  place  the  result  in  the  blank.  For  a  time  give  simple 
examples  and  require  the  use  of  the  formula.  Then  give  problems 
requiring  identification  of  data.  In  time  the  class  should  make  the 
statement  for  solution  as  soon  as  the  data  has  been  identified. 

VI.   GIVEN  THE  PERCENTAGE  AND  THE  RATE 
TO  FIND  THE  BASE. 

Example:     The  Percentage  is  $35,  the  Rate  5%,  find  the  Base. 
Formula  R      X      B    =      P 

Statement  of  relation  .05  x  $B  =  $35 
Statement  for  solution  $35  -5-  .05  =  $  Ans. 
After  giving  the  three  cases  in  Percentage,  emphasize  the  fact 
that  when  the  Percentage  is  wanted,  we  multiply;  when  the  Per- 
centage is  given,  it  is  divided  by  the  given  factor.  Dictate  many  ex- 
amples merely  to  be  stated  on  the  board  without  being  worked.  Dic- 
tation should  be  rapid,  but  not  so  rapid  but  that  all  members  of  the 
class  can  write  the  first  number  before  another  number  is  given. 
Unless  it  is  the  teacher's  fault,  do  not  repeat  a  number.  See  that 
every  member  of  the  class  writes  every  number  as  it  leaves  the 
teacher's  lips.  If  a  child  will  not  keep  up,  send  him  to  his  seat.  He 
will  soon  learn  not  to  lag  behind.  It  is  an  inexcusable  waste  of  time 
on  the  part  of  the  teacher  to  repeat  for  an  inattentive  pupil.  Such 
repetition  begets  inattention. 

Do  not  allow  the  pupils  to  hold  the  erasers  in  hand  and  erase 
when  they  please.  Leave  all  the  work  on  the  board  until  the  spaces 
are  well  filled,  then  say  "erase"  and  have  all  erase  at  the  same  time, 
thus  saving  valuable  time. 

63 


Many  examples  may  be  given  to  be  stated  without  solving.  The 
formula  and  statement  of  relation  should  be  written  and  the  state- 
ment for  solution  when  this  differs  from  the  statement  of  relation. 
The  blank  should  always  be  left  for  the  answer.  Many  simple  oral 
problems  should  be  given  to  the  class  at  their  seats,  also  many  diffi- 
cult problems  requiring  identification  of  data  should  be  read  from 
the  text  book,  the  numbers  identified  in  terms  of  percentage,  and  the 
method  of  solution  indicated  orally. 

A  point  that  should  be  brought  out  at  this  time  is  that  as  one 
whole  may  be  divided  into  several  parts,  so  one  Base  may  have  sev- 
eral Percentages  and  for  each  Percentage  there  is  a  rate. 

Example:  Mr.  Knox  had  $190  and  spent  $40  for  a  suit,  $35  for 
an  overcoat,  $5  for  a  hat,  and  $7  for  a  pair  of  shoes.  How  much  did 
he  have  left?  What  per  cent  had  he  left?  What  per  cent  did  the  suit 
cost?  The  overcoat?  Etc.  What  per  cent  did  he  spend  altogether? 
Prove  this  result  by  adding  the  per  cents  spent  for  the  several  items. 
Also  add  the  per  cents  spent  to  the  per  cent  left,  the  sum  being  100%. 

Example:  Mr.  Patterson  had  a  salary  of  $150  per  month.  He 
spent  20  %  for  rent,  8  lA  %  for  insurance,  26  2/3  %  for  household  ex- 
penses, 33  Yi  %  for  sundries,  and  saved  the  rest.  What  per  cent  did 
he  save?  What  was  his  rent?  Etc.  Check:  the  sum  of  the  several 
Percentages  should  equal  the  Base. 

VII.  AMOUNT. 

James  paid  40  cents  for  some  papers  and  sold  them  at  a  gain  of 
'J.0  cents.  What  did  he  sell  them  for?  This  we  call  the  Amount. 
What  is  the  40  cents  in  terms  of  percentage?  The  20  cents?  And 
what  is  the  60  cents  called?  Then  how  did  you  find  the  Amount? 
Who  will  define  Amount?  The  Base  plus  the  Percentage  equals  the 
Amount.  We  will  write  this  as  follows:  B  +  P  =  A,  and  call  it  the 
definition  of  Amount.  Have  children  commit  to  memory.  Give  many 
examples  to  find  the  Amount  when  the  Base  and  Percentage  are  given. 
Especially  emphasize  that  the  selling  price  when  there  is  a  gain  is 
the  Amount. 

When  we  have  given  the  sum  of  two  numbers  and  one  of  the 
numbers,  how  do  we  find  the  other  number?  If  12  is  the  sum  of  8  and 
some  other  number,  what  is  the  other  number?  Mr.  Ackerson  paid 
$150  for  a  horse  and  sold  it  for  $180,  what  did  he  gain?  What  do  we 
call  the  cost?  The  selling  price  when  there  is  a  gain?  The  gain? 
(The  teacher  should  emphasize  that  the  gain  is  always  the  Percent- 
age.) Then  how  did  we  find  the  Percentage  when  the  Base  and  the 
Amount  were  given?     Give  several  examples. 

Mr.  Hughes  sold  a  watch  for  $30,  thereby  gaining  $10,  What 
did  the  watch  cost?  What  was  the  $30?  The  $10?  The  20?  Then 
how  did  you  find  the  Base  when  the  Amount  and  Percentage  were 

64 


given?     Give  many  examples  illustrating  the  three  kinds  of  examples 
till  the  class  can  work  them  without  hesitation. 

VIII.  GIVEN  THE  AMOUNT  AM)  THE  BASE  TO  FIND 

THE  RATE. 

Emphasize  that  the  Rate  is  always  found  by  dividing  the  Per- 
centage by  the  Base,  therefore,  to  find  the  Rate  when  the  Base  and 
Amount  are  given,  find  the  missing  fact,  the  Percentage,  and  use  the 
formula  for  Percentage.  This  should  be  developed  by  means  of 
questions. 

IX.  GIVEN  THE  AMOUNT  AND  THE  PERCENTAGE 
TO  FIND  THE  RATE. 

By  questioning,  develop:  To  find  the  Rate  when  the  Amount 
and  Percentage  are  given,  find  the  missing  fact,  the  Base,  and  use  the 
formula  for  Percentage.  Give  many  examples  illustrating  the  last 
two  cases. 

X.  GIVEN  THE  PATE  AND  THE  BASE  TO  FIND  THE  AMOUNT. 

To  find  the  Amount  when  the  Rate  and  the  Base  are  given,  at 
first  have  the  class  solve  by  finding  the  missing  fact,  the  Percentage, 
then  add  this  to  the  Base.  But  it  is  an  advantage  to  be  able  to  solve 
by  multiplying  by  1  plus  the  Rate,  also  it  is  essential  to  an  under- 
standing of  the  next  case.  Before  teaching  the  following  lesson,  the 
fact  should  be  emphasized  that  any  number  is  100  r'<  of  itself.  $20 
is  1009^  of  $20.  100';  of  $75  is  $75.  The  purpose  of  the  following 
lesson  is  to  teach  the  formula:  (1.00  +  R)  xB  =  A. 
Preparation: 

RxB  =  P.  The  Base  equals  100  of  itself.  Base  plus  Percentage 
eouals  the  Amount. 

RXB  =  P. 

.05  x  $60  =  $3.        .06  X  $50  =  $3.        .07  X  $80  =  $5.60. 

$60  +  $3  =  $63.  A.     $50  +  $3  =  $53  A.     $80 +  $5.60  =  $85.60  A. 

Statement  of  aim:     We  are  now  going  to  learn  a  better  way  of 
finding  the  Amount  when  the  Rate  and  the  Base  are  given. 
Presentation: 

$60  is  what  per  cent  of  $60?  Ans.  100 'J  of  $60.  Teacher  writr 
onboard:  100 ';',  of  $60  =  $60.  $3  is  what  per  cent  of  $60?  Teacher 
write  on  board  below  last  statement:  5';  of  $60  =  $3.  How  many 
per  cent  of  $60  is  the  sum  of  100';  of  $60  and  59i  of  $60?  Ans. 
105%  of  $60.  Teacher  write  on  board  below  the  other  statements. 
How  many  dollars  is  the  sum  of  100';  of  $60  and  5';  of  $60?  Ans. 
$63.  Teacher  write  on  board.  We  now  have  the  following: 
100';  of  $60  =  $60 
of  $60  =  $  3 

Adding     105' ;    of  $60  =  $63 

G5 


Send  child  to  board  to  verify  this  statement  by  multiplying  $60 
by  1.05.  What  is  the  5%?  Ans.  The  Rate.  What  is  the  105%. 
Ans.  100%  plus  5%  or  100%  plus  the  Rate.  What  is  the  $60?  The 
$63?  Then  how  did  we  find  the  Amount?  or  What  did  we  multiply 
the  Base  by  to  find  the  Amount?  Ans.  100%  plus  the  Rate.  Similar- 
ly treat  the  other  two  examples  used  in  the  preparation. 

Comparison: 

In  all  these  examples  how  did  we  find  the  Amount?  Ans.  We 
multiplied  100%  plus  the  Rate  times  the  Base. 

Generalization: 

Give  me  a  rule  for  finding  the  Amount  when  the  Rate  and 
the  Base  are  given.  Rule:  To  find  the  Amount  when  the  Rate  and 
the  Base  are  given,  multiply  the  Base  by  100%  plus  the  Rate.  We 
may  write  this  as  a  formula  thus:      (1.00  +  R)  XB  =  A. 

This  we  shall  call  the  formula  for  Amount.  It  may  be  read 
either  100  %  plus  the  Rate  times  the  Base  equals  the  Amount,  or  I 
plus  the  Rate  times  the  Base  equals  the  Amount.  The  use  of  the 
curves  in  writing  the  formula  is  necessary.  See  that  the  class  dif- 
ferentiate between  the  formula  and  the  definition. 

Application: 

Send  class  to  board  to  write  and  interpret  the  formula.  Then 
dictate  examples  to  be  written  below  the  formula  in  statements.  At 
first  have  examples  stated  only,  then  have  some  of  them  worked. 
Example:     The  Base  is  $246,  the  Rate  is  8%,  state  for  Amount. 

(1.00  +  R)  XB  =  A. 

1.08  X  $246  =  $  Ans. 

The  addition  of  1.00  and  the  Rate  should  always  be  made  men- 
tally and  the  example  stated  as  above  at  once  as  the  teacher  dictates. 
Assignment: 
Problems  from  the  text  book  should  be  read  and  the  numbers 
identified  in  terms  of  percentage  and  the  method  of  solution  indicat- 
ed.    They  should  then  be  worked  during  the  study  period. 


XI.  GIVEN  THE  AMOUNT  AND  THE  RATE  TO  FIND  THE  BASE 

Example:  The  Amount  is  $428,  the  Rate  7%,  state  for  finding 
the  Base. 

Formula  (1.00  +  R)XB=A 

Statement  of  relation  1.07X$B  =  $428 

Statement  for  solution     $42S-h1.07=$         Ans. 

Upon  reaching  this  point  the  majority  of  a  class  will  work  this 
example  without  help  from  the  teacher  except  the  suggestion  that  the 
formula  for  Amount  be  used  and  the  numbers  placed  under  the  ap- 
propriate letters. 

66 


XII.  GIVEN  THE  AMOUNT  AND  THE  RATE  TO  FIND 

THE  PERCENTAGE. 

Develop:     Find  the  essential  missing  fact,  the  Base,  and  subtract. 

XIII.  GIVEN  THE  PERCENTAGE  AND  RATE  TO  FIND 

THE  AMOUNT. 

Develop:     Find  the  essential  missing  fact,  the  Base,  and  add. 

Many  miscellaneous  problems  applying  all  the  cases  in  percent- 
age thus  far  covered  should  now  be  given  for  identification  of  data, 
for  statement,  or  for  solution.  Also  give  frequent  special  drills  in 
identification  of  data  as  follows:  Send  class  to  the  board  to  write 
the  appropriate  letter  as  the  teacher  dictates;  the  whole,  the  part, 
the  cost,  the  gain,  the  selling  price  when  there  is  a  gain,  what  you 
have  to  start  with,  etc.  Teacher  should  dictate  rapidly  and  then  have 
all  errors  checked.     The  order  of  dictation  should  vary. 

XIV.   DIFFERENCE. 

A  dealer  paid  $12  for  some  goods  and  then  on  account  of  their 
being  damaged  sold  them  at  a  loss  of  $3,  what  was  the  selling  price? 
What  is  the  $12?  The  $3?  Then  how  did  you  find  the  $9?  Ans.  We 
subtracted  the  Percentage  from  the  Base.  We  call  this  $9  the  Dif- 
ference because  it  is  obtained  by  subtraction.  Define  Difference. 
The  Base  minus  the  Percentage  equals  the  Difference.  We  write  this 
thus:  B — P  =  D,  and  call  it  the  definition  of  Difference.  If  the  Base 
is  $74  and  the  Percentage  $14,  what  is  the  Difference?  Etc.  Empha- 
sis should  be  laid  on  the  fact  that  the  selling  price  when  there  is  a 
loss  is  the  Difference. 

Mr.  Jerome  paid  $35  for  vegetables  and  sold  them  for  $30,  how 
much  did  he  lose?  What  is  the  $35?  The  $30?  The  $5?  Then  how 
did  you  find  the  Percentage  when  the  Base  and  the  Difference  were 
given?     Etc. 

Frank  sold  his  sister  a  tablet  for  $.10  thereby  losing  $.05,  what 
did  the  tablet  cost  him?  What  is  the  $.10?  The  $.05?  The  $.15? 
Then  how  did  you  find  the  Base  when  the  Difference  and  Percentage 
were  given?  Etc.  The  class  should  become  proficient  in  the  three 
kinds  of  examples  given  above. 

XV.  GIVEN  THE  DIFFERENCE  AND  THE  BASE  TO  FIND 

THE  RATE. 

Develop:  Find  the  essential  missing  fact,  the  Percentage,  then 
find  the  Rate  in  the  usual  way. 

XVI.  GIVEN  THE   DIFFERENCE  AND  THE  PERCENTAGE 

TO  FIND  THE  RATE. 

Develop:     Find  the  essential  missing  fact,  the  Base,  again  em- 

67 


phasizing  that  the  Percentage  is  always  divided  by  the  Base  to  find 
the  Rate. 

XVII.  GIVEN  THE  RATE  AND  THE  BASE  TO  FIND 
THE  DIFFERENCE. 

To  find  the  Difference  when  the  Rate  and  the  Base  are  given,  at 
first  have  the  class  solve  by  finding  the  missing  fact,  the  Percentage, 
then  subtract  this  from  the  Base.  Later  teach  the  following  formula: 
(1.00 — R)xB  =  D.  See  that  the  class  differentiate  between  the 
formula  and  the  definition. 

This  lesson  should  be  presented  as  the  corresponding  lesson  in 
Amount  was  presented.  The  examples  in  the  preparation  would  be 
like  the  following: 

RX      B=   P 
.05x$60  =  $  3 
$60—$  3  =  $57  D 
The  questions  in  the  presentation  should  lead  to  the  following: 
100%   of  $60  =  $60 
5%   of  $60  =  $  3 

Subtracting,  95%  of  $60  =  $57. 
Verify  and  bring  out  that  to  find  the  Difference  we  multiplied 
the  Base  by  100 '/r  minus  the  Rate.  In  the  application  have  class 
write  formula  and  statement,  subtracting  the  Rate  from  1.00  men- 
tally. Example:  The  Base  is  $380,  the  Rate  12%,  fiind  the  Differ- 
ence. 

(1.00— R)  XB  =  D 

.88  X  $380  =  $         Ans. 

XVIII.  GIVEN  THE  DIFFERENCE  AND  THE  RATE  TO  FIND 

THE  BASE. 

Example:  The  Difference  is  $399.50,  the  Rate  15%,  state  for 
finding  the  Base. 

Formula  (1.00— R)X   B  =  D 

Statement  of  relation  .85X$B=$399.50 

Statement  of  solution     $399.50-=-. 83=$  Ans. 

XIX.  GIVEN  THE  DIFFERENCE  AND  THE  RATE  TO  FIND 

THE  PERCENTAGE. 

Develop:     Find  the  essential  missing  fact,  the  Base  and  subtract. 

XX.  GIVEN  THE  PERCENTAGE  AND  THE  RATE  TO  FIND 

THE  DIFFERENCE. 

Find  the  essential  missing  fact,  the  Base,  and  subtract. 

68 


Examples  in  Profit  and  Loss,  Commission,  and  Cash  Discount 
should  be  given  to  apply  all  the  cases  in  percentage.  Continue  the 
drill  in  identification  by  writing  the  appropriate  letter  on  the  board 
as  the  teacher  dictates.  Now  the  terms  dictated  may  include:  what 
you  start  with,  the  whole,  the  cost,  the  list  or  marked  price,  the  sell- 
ing price  in  commission,  the  principal,  all  of  which  are  the  Base;  the 
part,  the  gain,  the  loss,  the  commission,  the  discount,  all  of  which  are 
the  Percentage;  the  selling  price  when  there  is  a  gain,  the  sum  sent 
to  an  agent  to  pay  for  goods  and  commission,  any  sum  after  a  gain. 
Amount;  the  part  left,  the  selling  price  when  there  is  a  loss,  the 
selling  price  when  there  is  a  discount,  the  net  proceeds  of  a  sale  on 
commission  when  there  are  no  extra  expenses,  Difference.  When  the 
Rate  is  called  for,  have  class  state  what  Rate;  as,  Rate  of  commis- 
sion, Rate  of  gain,  etc.  Emphasize  that  the  word  after  "of  follow- 
ing "Rate"  indicates  the  Percentage.  When  the  Rate  is  the  Rate  of 
commission,  the  Percentage  is  the  commission;  when  the  Percentage 
is  the  gain,  the  Rate  is  the  Rate  of  gain;  etc.  Emphasize  that  the 
selling  price  may  be  either  Base,  Amount,  or  Difference,  bringing  out 
that  it  must  be  known  whether  goods  are  sold  at  a  loss  or  a  gain,  on 
commission,  or  at  a  discount,  before  the  selling  price  can  be  identified. 
Also  after  calling  attention  to  the  fact  that  in  profit  and  loss  the  cost 
is  always  the  Base,  bring  out  that  in  discount  the  Difference,  which 
is  the  selling  price  to  the  man  selling  the  goods,  is  the  cost  to  the 
man  buying. 

In  connection  with  the  work  in  percentage,  the  teacher  should 
review  the  section  on  "Problem  Solving";  and  when  the  class  are 
writing  down  the  identification  of  data  either  on  paper  or  the  board, 
they  should  always  indicate  what  Rate  is  given  or  wanted;  as.  the 
"Rate  of  loss"  or  the  "Rate  of  Comm.",  not  merely  the  "Rate".  This 
often  is  a  material  aid  in  identifying  the  Percentage  itself. 

XXI.  SEVERAL  SUCCESSIVE  DISCOUNTS. 

Example:  The  list  price  of  a  piano  sold  by  Potter  and  Putnam 
is  $600  with  10%,  8#,  and  5$    off.     What  is  the  selling  price? 

The  example  may  be  worked  in  either  of  two  ways.  Ordinarily 
the  quickest  way  is  to  use  the  following  formula: 

(1.00— Ri  )  (1.00— R2  >  (1.00— R3  )  etc.)XB=S.P. 

The  child  should  subtract  the  Rates  mentally,  stating  the  exam- 
ple at  once: 

.90  X  .92  X  .95  X  $600  =  $471.96.     Ans. 

This  method  certainly  has  the  advantage  of  compactness  of 
statement,  while  in  the  number  of  figures  required  in  the  operations 
there  is  also  a  saving.  The  child  working  at  the  board  to  the  teach- 
er's dictation  can  state  and  work  the  example  by  this  method  in  half 
the  time  required  by  the  method  commonly  in  use.  As  the  teacher 
dictates,  the  child  makes  the  mental  subtractions  and  has  the  example 

69 


stated  as  soon  as  dictation  ceases.  All  that  remains  to  be  done  is  to 
make  the  abstract  computations  and  place  the  answer  in  the  state- 
ment. The  constant  oscillation  between  statement  and  work  is 
avoided.  Give  much  practice  in  merely  stating  such  problems  with- 
out solving  them. 

XXII.  TO  FIND  A  SINGLE  RATE  OF  DISCOUNT  EQUIVALENT 
TO  SEVERAL  SUCCESSIVE  RATES. 

Use  the  following  formula: 

1.00— (1.00  -Ri  )  (1.00— Ra  )  <etc.)  =  R 
Example:     What  single  Rate  of  discount  is  equivalent  to  8% 

and  10%  off? 

1.00— .92  X.  90  =  11%%.     Ans. 

XXIII.  MARKING  GOODS. 

Merchants  sometimes  wish  to  mark  goods  for  a  sale  so  that  they 
may  be  able  to  make  a  profit  on  the  cost  of  the  goods  but  at  the  same 
time  be  able  to  offer  a  discount  on  the  marked  price.  For  example,  a 
dealer  wishes  to  sell  goods  that  cost  $240  so  that  he  will  make  a  profit 
of  25%  on  this  cost,  but  still  be  able  to  discount  his  marked  price 
20%.  To  make  25%,  he  must  sell  for  $300.  He  marks  his  goods 
$375  and  sells  at  a  discount  of  20%  on  this  marked  price,  thus  re- 
ceiving the  $300  he  wishes.  This  involves  a  change  of  Base,  the  Base 
being  the  cost  in  profit  and  loss,  and  the  marked  price  in  discount. 
The  selling  price  is  the  Amount  in  profit,  and  the  Difference  in  dis- 
count, therefore  the  $300  is  the  Amount  in  one  case  and  the  Differ- 
ence in  the  other.  We  multiply  the  Base  $240  by  1.25  to  obtain  the 
selling  price,  then  divide  this  selling  price  as  the  Difference  in  dis- 
count by  1.00— R,  or  .80,  giving  us  the  other  Base,  the  marked  price. 
That  is,  (1.00 +  R,  )  X  B  is  the  selling  price,  Rt  being  the  Rate  of 
gain  and  B  the  cost.  This  selling  price  as  the  Difference  divided  by 
(1.00 — R2 )  gives  us  the  other  Base,  the  marked  price,  R2  being  the 
Rate  of  discount.     Expressing  this  as  a  formula,  we  have: 

(1.00  +  Ri  )XB 
=  Marked  Price. 


1.00— R2 
As  the  teacher  dictates  the  problem,   the  class  should  add  and 
subtract  the  Rates  mentally  and  state  the  problem  promptly  as  fol- 
lows: 

1.25X$300 

=  $  Ans. 

.80 

As  usual  the  teacher  should     give     problems  to  be  stated  only, 
sometimes  giving  the  Rate  of  discount  before  the  Rate  of  gain. 

70 


If  the  cost  is  not  stated,  and  the  question  is  how  must  goods  be 
marked  to  gain  25 ',',  and  still  allow  a  discout  of  20 r  J  ?  assume  $1.00 
as  the  cost. 

1  ^5X$1 

=$1.56  1/4. -.561  1    above  coat 

.80 

But  as  1  as  a  factor  does  not  change  the  result,  the  $1  may  be 
omitted  and  the  example  stated  as  follows: 

1.25 
=1.56  1  4.  .56  1  4V  above  cost. 


.80 
Put  as  a  formula  this  would  be  either: 

1.00  + Ri 

=  M.  P.  in  per  cent. 

1.00— R2 

Or  if  the  formula  is  to  express  the  Rate  above  cost: 

1.00-t-Rr 

—  1  =  R  above  cost. 


1.00— Ra 

Example:  How  must  pianos  costing  $850,  $500,  and  $300  be 
marked  so  that  the  dealer  may  give  a  discount  of  20';  and  still  gain 
25%? 

1.25/.80  =  1.56J4.        1.56J4  X$500  =  $     Ans.  1.56 '^  x  $850  = 

$  Ans.  1.56  ?4  X$300  =  $  Ans.  It  will  be  seen  that  the 

advantage  of  this  method  is  that  the  dealer  gets  his  Rate  once  for  all 
articles,  and  then  mu'tiplies  each  cost  by  this  Rate,  instead  of  multi- 
plying the  cost  of  each  article  by  1.25  and  dividing  each  result  by  .80. 

XXIV.   SIMPLE  INTEREST. 

As  a  preparation  for  Simple  Interest,  teach  reduction  of  months 
and  days  to  years  using  the  following  progressive  order:  months  to 
years;  years  and  months  to  years;  days  to  years;  years  and  days  to 
years;  months  to  days;  months  and  days  to  days  and  then  to  years; 
years,  months  and  days  to  years,  first  reducing  the  months  and  days  to 
days.  Four  months  equals  4/12  of  a  year  equals  1/3  of  a  year;  3 
years,  4  months  equals  10/3  years;  20  days  equals  20/360  of  a  year 
equals  1/18  of  a  year;  4  years,  18  days  equals  4  1/20  years  equals 
81/20  years;  3  months,  10  days  equals  100  days  equals  5/18  of  a 
year;  2  years,  3  months,  10  days  equals  2  5/18  years  equals  41  L8 
years. 

Interest,  which  is  money  paid  for  the  use  of  mony  or  capital,  is 
paid  annually,  hence  we  have  a  new  element,  Time.  What  will  be  the 
Interest  for  one  year  on  $450  at  5'/,  ? 

RX      B  =  P 

. 05 X $450 =$22. 50  Interest. 

71 


What  will  be  the  Interest  for  2  years?     3  years?      Vz  year? 

2 X. 05 X $450. 00  =  $45. 00   Interest  for  2  years. 

3X.05X$450.00=$67.50    Interest  for  3  years. 

1  2X.05X$450.00=$11.25    Interest  for  12  year. 

Letting  I  stand  for  Interest  and  T  for  Time  in  years,  we  have  in 
general:  TxRxB  =  I.  What  will  be  the  Interest  on  $300  for  3  yrs. 
4  mos.  at  6rA  ?      (Use  cancellation.) 

T    X  R  X  B    =1 
10/3X.06X$300=$60.     Ans. 

What  will  be  the  Interest  on  $600  for  2  years,  12  days  at  5%? 

In  computing  Interest,  one  year  is  now  generally  treated  as  360 
days,  and  one  month  as  30  days;  therefore  12  days  equals  12/360  of 
a  year,  or  1/30  yr. 

61/30  X   .05  X  $600  =  $61.     Ans. 

Find  the  Interest  on  $200  for  9  mo.  10  da.  at  3%. 

280/360  =  7/9.      7/9X  .03  X  $200  =  $4.66%.\$4.67.      Ans. 

Find  the  Interest  on  $115  for  2  yrs.,  1  mo.,  10  da.,  at  5%. 

19/9X.05X$H5  =  $12.14.     Ans. 


XXV.  GIVEN  THE  INTEREST  AND  TWO  ELEMENTS  TO  FIND 

THE  THIRD  ELEMENT. 

Preparation:     2x3x5  =  30.      30- (2x5)  =3;     30-5-  (2x3)  =5; 
30-M3x5)=2.     TxRxB  =  I. 

Example:     In  how  long  a  time  will  $470  placed  at  4%  Interest 
produce  $65.80  Interest? 

T  X. 04  X  $470  =  $65.80. 

$65.80-^-(.04x$470)=3I/>-'-3  yr.,  6  mo.     Ans. 
If  the  result  were  2  7/9  years,  7/9  of  360  days  is  280  days  which 
equals  9  months  10  days,  giving  2  years,  9  months,  10  days. 

Example:     At  what  rate  will  $240  produce  $32.40  interest  in  2 
years,  3  months? 

$32.40-5-  (2 %  X  $240)  =.06.     .-.6%     Ans. 
Example:    What  principle  placed  at  5%  interest  for  4  years  will 
produce  $28.80  interest? 

$28.80- (4 X. 05)  =$144.     Ans. 


XXVI.  SIX  PER  CENT  METHOD. 

The  teacher  should  develop  the  method  of  subtracting  dates  and 
of  finding  the  Rate  at  6%  for  the  given  time.  For  one  month  the 
Rate  would  be  one  twelfth  of  .06  or  .005;  for  one  day  one  thirtieth 
of  .005  or  .000^  or  .001/6. 

I.     Find  the  Time  by  subtracting  dates  if  necessary. 

72 


II.   Find  the  Total  Rate  for  the  given  time: 
Number  of  years       X     06 
Number  of  months    X    .005      = 
Number  of  days        X  .000  '  /,     = 


Total   Rate 

III.  Find  the  Interest: 

r/6x  Total  RxB  =  I. 

IV.  Find  the  Amount: 

B  +  I  =  A. 
Multiplying  the  principal  by  the  Total  Rate  for  the  given  time 
would  give  the  Interest  at  §'/<  .  Dividing  this  by  6  gives  the  Interest 
at  1%;  and  multiplying  the  Interest  at  1%  by  r  (not  by  R),  gi 
the  Interest  at  the  given  Rate.  Here  r  equals  the  number  of  per 
cent.  For  example:  if  the  Rate  is  5',',  ,  R  =  .05  and  r=5.  In  practice 
we  do  not  actually  find  the  Interest  at  6 'A  and  then  divide  by  6  and 
multiply  by  r.  We  at  once  cancel  the  6  when  possible  and  then  multi- 
ply in  any  order  deemed  shortest.  Have  the  class  place  blank  on  the 
board  before  beginning  dictation.  Insist  on  the  vertical  arrangement 
and  correct  relative  position  of  all  work  by  all  members  of  class.  In 
the  end  this  is  a  time  saving  device. 

SIX  PER  CENT  BLANK. 


X 

.06 

= 

. 

X 

.005 

= 

. 

X 

.000% 

= 

• 

$ 

X    $ 

= 

$ 

$ 

Int. 
Amt. 

Ans. 
Ans 

r/6x 

$  + 

Example:     Find  the  Interest  and  Amount  at  7rf   for  a  note  for 
$360,  dated  August  27,  1905  and  paid  December  8.  1911. 
1911— 12--8     6X.06        =.36. 
1905—8-27     3X.005      =.015 

11  X.  000%=.  001  % 

6 — 3-11  


.376-% 
7/6  X. 376 •%  X  $360  =  $158.27     Int.      Ans. 
$158.27  +  $360  =  $518.27     Amt.     Ans. 

Give  much  rapid  oral  work  in  the  solution  of  problems  in  Interest. 
As  the  Rate  at  6%  for  60  days  is  .01,  to  find  the  Interest  for  60  days, 
move  the  decimal  point  two  places  to  the  left;  to  find  the  interest  for 
30  days,  move  the  point  two  places  to  the  left  and  divide  by  2;  to 
find  the  Interest  for  120  days,  move  the  point  and  multiply  by  2.  For 
other  terms,  various  combinations  of  the  above  method  may  be  used. 

73 


The  Interest  on  $240  for  60  days  at  6%  is  $2.40;  for  30  days,  $1.20; 
for  90  days,  $3.60;  for  120  days,  $4.80.  To  find  the  Interest  at  5%, 
find  H  of  the  Interest  at  6%. 

XXVII.  COMPOUND  INTEREST. 

Compound  Interest,  though  it  is  illegal  for  general  use,  is  still 
employed  by  saving  banks,  which  generally  compute  Interest  semi- 
annually and  add  this  to  the  principal  for  a  new  principal  upon  which 
the  Interest  for  the  next  term  is  computed.  Letting  R  equal  the 
Rate  for  the  term  of  Interest,  and  t  the  number  of  terms,  develop  the 
formula  below: 

Example:  Find  the  Amount  of  $300  compounded  annually  for 
3  years  at  4%. 

Example:  Find  the  Interest  on  $400  compounded  semi-annually 
for  3  years  at  4%. 

(1.00  +  RH  XB=     A 
1.04  3x$300=$347.46 
1.026  X$400=$450.46 
$450.46— $400=$50.46     Compound  Int.   Ans. 

Several  simple  examples  should  be  worked  by  actually  multiply- 
ing them  out,  as  in  the  first  example  above  multiplying  1.04  by  itself 
three  times  and  $300  by  the  result.  Then  more  difficult  example' 
like  the  second  should  be  worked  by  use  of  the  following  table.  State 
the  example  exactly  as  above,  then  in  the  table  in  the  column  for  2% 
and  opposite  the  year  6  will  be  found  the  Amount  of  $1.00  at  2%  for 
6  years,  which  is  the  same  as  the  Amount  of  $1.00  for  3  years  at  4% 
compounded  semi-annually.  Multiplying  this  $1.12616  by  400  gives 
the  desired  result,  $450.46. 


;  showing  the  amount  of  $1  at  compound 

interest  for 

Year 

2  Per  Cent 

2%  Per  Cent 

3  Per  Cent  &£  Per  Cent 

4  Per  Cent 

1 

1.02000 

1.02500 

1.03000 

1.03500 

1.04000 

2 

1.04040 

1.05063 

1.06090 

1.07123 

1.08160 

3 

1.06121 

1.07689 

1.09273 

1.10872 

1.12486 

4 

1.08243 

1.10381 

1.12551 

1.14752 

1.16986 

5 

1.10408 

1.13141 

1.15927 

1.18769 

1.21665 

6 

1.12616 

1.15969 

1.19405 

1.22926 

1.26532 

7 

1.14869 

1.18869 

1.22987 

1.27228 

1.31593 

8 

1.17166 

1.21840 

1.26677 

1.31681 

1.36857 

9 

1.19509 

1.24886 

1.30477 

1.36290 

1.42331 

10 

1.21899 

1.28009 

1.34392 

1.41060 

1.48024 

XXVIII.  BANK  DISCOUNT. 

As  an  introduction  to  the  subject  of  Bank  Discount,  the  teacher 

74 


should  explain  to  the  class  the  two  chief  functions  of  banking:  deposit 
and  loan  or  discount.  She  should  provide  herself  through  some  local 
hank  with  the  various  blank  forms  used  in  banking:  the  pass  book, 
the  deposit  slip,  the  check  book  including  checks  and  stubs,  the  bank 
note,  and  the  draft. 

All  banks  receive  money  on  deposit  for  safe  keeping,  sometimes 
allowing  Interest,  sometimes  not.  Saving  banks,  which  do  not  accept 
deposits  for  checking  purposes,  allow  Interest  on  all  sums  left  on  de- 
posit a  stated  length  of  time.  Commercial  banks  allow  Interest  on 
sums  left  on  deposit  without  being  checked  upon.  When  money  [a 
deposited  merely  for  checking  purposes,  in  some  cases  no  Interest  is 
allowed;  in  others,  Interest  is  allowed  on  unchecked  balances. 

The  teacher  should  further  explain  the  method  and  purposes  of 
checking;  how  personal  accounts  between  individuals  or  firms  are 
balanced  by  the  use  of  checks  instead  of  money.  It  is  well  to  explain 
how  such  checks  often  pass  from  hand  to  hand,  especially  between 
banks,  in  the  payment  of  debts  or  the  settlement  of  accounts,  before 
they  are  finally  presented  for  payment  at  the  bank  upon  which  they 
are  drawn.  In  this  connection  the  function  of  the  New  York  Clearing 
House  should  be  explained. 

Review  Simple  Interest  and  show  that  Bank  Discount  is  nothing 
more  nor  less  than  Simple  Interest  paid  in  advance  by  the  borrower 
to  the  bank  for  the  use  of  the  bank's  money.  Discounting  notes  is  the 
bank's  method  of  loaning.  When  a  person  of  good  financial  standing. 
or  one  who  can  give  good  security  for  the  repayment  of  the  money, 
wishes  to  borrow,  say  $400,  from  the  bank;  he  draws  up  a  promisory 
note  payable  to  himself  or  another  at  the  bank  in  30,  60,  90,  or  120 
days.  The  bank  generally  charges  for  this  loan  the  legal  rate  of  in- 
terest for  the  given  time  and  collects  this  charge  in  advance.  Sup- 
pose the  time  to  be  60  days,  the  Interest  charge,  or  the  Bank  Discount 
will  be  $4.  Thus  the  bank  loans  the  borrower  $400  and  receives  in 
return  the  promisory  note  and  $4  for  the  use  of  the  loan.  Or  what  is 
the  same  thing,  the  bank  deducts  the  $4  from  the  $400  and  pays  out 
the  balance,  $396,  called  the  Proceeds.  Such  examples  differ  from 
Simple  Interest  only  in  that  the  Interest  is  subtracted  to  find  the 
Proceeds  instead  of  being  added  to  find  the  Amount  due  when  the 
note  is  payable.  If  the  borrower  in  the  above  case  had  wanted  the 
use  of  the  money  for  more  than  120  days,  instead  of  making  out  the 
note  for  a  longer  time,  he  would  have  it  renewed  when  it  became  due 
and  would  again  discount  the  $400  for  the  additional  time. 

Frequently  business  men  receive  from  their  customers  in  pay- 
ment of  accounts  notes  payable  when  the  maker's  crops  will  mature. 
If  such  notes  are  long  time  interest  bearing  notes,  they  are  often 
deposited  at  a  bank  for  collection,  in  which  case  no  account  is  made 
of  the  notes  on  the  books  of  the  bank,  except  to  give  the  payee  credit 
when  the  notes  are  paid.     When,  however,  these  notes  are  short  time 

75 


non-interest  bearing  notes,  the  payee  may  in  case  of  need  of  ready 
money  have  them  discounted  at  the  bank  at  any  time  before  they  are 
due.  In  such  case,  the  discount  is  computed  for  the  number  of  days 
the  note  has  yet  to  run  and  upon  the  amount  that  will  be  due  when 
the  note  matures.  In  the  solution  of  such  problems,  the  use  of  the 
following  blank  will  be  found  advantageous: 


Date 
Discounted 


BANK  DISCOUNT  BLANK. 

Due 


t     = 


Due      

,'6X(         X.001/6)X$  =$  B  Dis.     Ans.  t 

$  _$  =$  Pro.     Ans. 

When  the  term  is  30,  60,  90,  or  120  days,  the  appropriate  rate 
should  be  placed  above  the  parenthesis  at  once  without  multiplication. 

Example:     Find  the  Bank  Discount  and  the  Proceeds  of  a  $120 

non  interest  bearing  note  dated  June  3  and  due  in  90  days,  if  dis- 
counted at  the  bank  July  12  at  5%. 

90  June  27 

Date— June  3~  I   t  _  M  !   Due— Sept.  1  July  31—19 

J  ;  ;  Aug.  31— 31 

Discounted— July  12       Due    Sept.   1—  1 

90     51  t 

5/6X(5lX.001/6)X$I20=$.85.     B.  Dis.     Ans. 
$120— $.85  =$119.15.     Pro.     Ans. 

Before  giving  problems  to  find  the  Bank  Discount,  it  would  be 
well  to  give  drill  in  finding  the  date  due  and  the  term  of  discount. 
First  make  a  quick  estimate  as  to  date  due.  In  the  above  example 
the  date  due  must  be  about  Sept.  3.  If  in  a  90  day  note  the  date  due 
as  finally  found  differs  more  than  two  days  from  the  estimated  date, 
some  error  must  have  been  made.  To  find  the  exact  date,  place  the 
number  of  days  the  note  is  to  run,  90,  below  the  column;  after  "June" 
the  number  of  days  after  June  3;  after  "July"  and  "Aug.",  the  num- 
ber of  days  in  those  months  respectively,  and  after  "Sept."  the  num- 
ber necessary  to  make  the  total  90.  Do  not  add  and  then  subtract 
from  90,  but  adding  say:  9  and  1  are  10,  placing  the  1  in  the  blank 
opposite  "Sept."  Continuing  say:  3 — 9.  Thus  getting  the  date 
Sept.  1. 

To  find  the  term  of  discount,  in  the  second  column  after  "July" 
place  19,  the  number  of  days  after  July  12,  the  date  of  discount. 
After  the  remaining  months,  July  and  Aug.,  place  the  same  numbers 
as  in  the  first  column.     Add  and  the  result  is  the  term  of  discount. 

76 


Before  even  this  work  can  be  done  accurately,  each  child  must. 
have  at  tongue's  end  the  number  of  days  in  each  month. 

As  the  Rate  of  Discount  at  6fA  for  one  day  is  .001/6,  to  find  the 
Discount  for  a  given  number  of  days,  move  the  point  three  places  to 
the  left,  divide  by  6,  and  multiply  by  the  given  number  of  days.  Find 
the  discount  on  $1200  at  G<'<  for  17  days.  Mentally  moving  the  point 
three  places  to  the  left  and  dividing  by  6  gives  $.20,  the  Discount  for 
one  day,  and  multiplying  this  by  17  gives  $3.40,  the  Discount  for  17 
days. 

XXIX.  DISCOUNTING  INTEREST  BEARING  NOTES. 

As  Discount  is  computed  upon  the  sum  due  at  maturity,  the 
Amount  of  an  interest  bearing  note  instead  of  the  face  would  be  the 
Base  in  Bank  Discount;  therefore  find  the  Amount  by  the  6' ',  method 
and  use  this  Amount  as  the  Base  in  the  Bank  Discount  blank. 

Example:  Find  the  Proceeds  of  a  90  day  note  for  $400  bearing 
Interest  at  5%,  dated  Jan.  10,  1912,  and  discounted  Mar.  5  at  6'.  . 

90  Jan.      21 

Date— Jan.  10  1912  ~ 


_  ls;  |    Due— Apr. 9.         Feby.  29 

*  -  5*  i                                    Mar.    31—26 

Discounted— Mar.  5  Due   Apr.      9 —  9 

5'6X.005X$400  =$5  I. 

$5  +$400  =$405  A. 

6  6X(35X.001  6)X$405  =$     2.36  B.  Dis. 


$402.64  Pro,  Ans. 


77 


ADDENDA. 


PREVENTION  OF  COPYING. 

1.  Remove  the  possibility  of  copying  by  seating  the  class  in  a 
manner  to  prevent  it,  and  by  giving  different  examples  to  neighbor- 
ing children  at  the  board.  On  tests  with  all  teachers  and  at  all  times 
with  inexperienced  teachers  and  generally  in  the  lower  grades,  this 
method  is  advisable,  but  it  does  not  get  at  the  root  of  the  evil.  The 
tendency  to  copy  remains  and  passive  attention  is  relied  upon,  where- 
as the  power  of  active  attention  with  effort  should  be  developed.  To 
accomplish  this  end,  the  teacher  must  be  constantly  on  the  alert  and 
she  must  employ  various  means  to  prevent  copying  when  the  possi- 
bility and  temptation  are  always  present. 

2.  Inflict  some  penalty,  (a)  Turn  the  erring  child  around  with 
his  back  to  the  board  till  the  rest  of  the  class  finish  the  example, 
(b)  Mark  him  zero  for  the  example,  (c)  Send  him  to  his  seat.  But 
all  these  appeal  to  a  low  motive  and  the  child  will  still  copy  when 
sure  of  avoiding  the  penalty.  However  he  is  given  some  discipline  in 
that  he  inhibits  a  desire  to  look  at  his  neighbor's  work  and  thus 
strengthens  his  active  attention. 

3.  Appeal  to  his  sense  of  shame  or  pride  or  better  his  self  res- 
pect by  making  him  see  how  other  people  regard  such  actions.  Here 
again  he  will  continue  to  copy  when  he  thinks  he  is  sure  of  avoiding 
detection. 

4.  Show  the  child  the  practical  results  of  copying,  that  it  pre- 
vents his  understanding  the  subject  and  keeps  the  teacher  from 
knowing  when  to  give  him  proper  help  and  hence  he  will  probably 
fail  and  remain  in  the  grade  another  term,  or  that  possibly  he  will  be 
demoted  to  a  lower  grade.  This  appeal  too  will  be  of  only  temporary 
value,  as  when  he  comes  to  take  a  test,  he  knows  that  the  practical 
result  is  the  other  way  around. 

5.  Make  the  child  see  that  copying  prevents  mental  development, 
that  instead  of  gaining  mental  power  he  is  weakening  his  ability  to 
think  in  the  future.  This  will  appeal  to  the  ambitious  child  but  the 
child  who  thinks  only  of  the  immediate  present  will  receive  litt'e 
stimulus. 

All  of  these  fail  in  the  end  unless  reinforced  by  some  motive  that 
applies  with  equal  force  to  all  possible  cases.  Some  of  these  means 
will  be  found  effective  under  some  circumstances  and  ineffective  un- 
der others.  By  these  the  child  who  cannot  be  reached  by  a  higher 
appeal  must  be  reached. 

6.  Appeal  to  the  moral  sense.  It  is  not  necessary  to  paint  copy- 
ing in  black  letters  and  brand  it  as  cheating  and  stealing.  Many 
children  do  not  have  sufficient  moral  s^nse  to  realize  that  it  is  wrong, 

78 


and  the  use  of  strong  language  is  apt  to  do  more  harm  than  good. 
If  possible,  however,  the  child  should  be  made  to  see  the  evil,  that  it 
is  really  wrong,  that  it  is  unfair  to  the  rest  of  the  class.  How  to  ac- 
complish this  will  depend  upon  the  child  and  upon  circumstances. 
Sometimes  the  appeal  should  be  to  the  whole  class  and  at  times  to  the 
child  privately.  Many  children  if  convinced  of  the  wrong  will  cease 
the  evil  practice,  while  of  course  others  will  deliberately  continue  to 
copy  knowing  it  to  be  wrong.  With  the  latter  some  of  the  other 
means  discussed  will  be  essential. 

ONLY  ONE  DIFFICULTY  AT  A  TIME. 

Under  the  discussion  regarding  teaching  a  new  operation,  em- 
phasis was  laid  upon  introducing  but  one  new  difficulty  at  a  time.  A 
few  specific  cases  will  here  be  given. 

In  column  addition,  which  should  be  begun  as  soon  as  the  child 
knows  the  facts  sum  seven,  the  new  difficulty  is  remembering  the  sum 
of  the  first  two  digits  and  adding  this  to  the  third  digit,  with  but  one 
of  the  two  numbers  in  sight.  As  a  preparation  for  this,  it  might  be 
well  to  place  some  number,  as  2,  on  the  board  and  then  have  the  class 
add  to  this  any  number  from  1  to  5  dictated  by  the  teacher.  Suppose 
the  column  to  be  2-2-3,  ask  the  child  the  sum  of  the  first  two  digits, 
then  ask  him  the  sum  of  this  number  and  the  last  digit.  Focus  the 
child's  mind  upon  the  thought  that  he  is  to  add  the  sum  of  the  first 
two  digits  to  the  third,  thus  making  the  new  idea  as  vivid  as  possible, 
the  strong  first  impression.  At  first  the  work  will  have  to  be  oral, 
the  teacher  questioning  and  the  child  answering,  then  the  child  doing 
all  the  oral  work  alone,  and  finally  adding  mentally  and  giving  the 
final  answer  only. 

The  next  difficulty  in  column  addition  is  carrying  and  here  again 
the  child's  mind  must  be  focussed  upon  whether  there  is  a  carry  and 
that  he  must  add  it  as  soon  as  he  determines  that  there  is.  Of  course 
if  any  of  the  addition  facts  used  are  not  thoroughly  known,  the  new 
difficulty  will  not  be  the  only  one  that  the  child  must  contend  with, 
the  attention  will  be  distributed,  and  hence  the  first  impression  will 
not  be  as  vivid  as  it  would  be  if  the  attention  could  be  focussed  upon 
the  one  new  idea.  This  emphasizes  the  importance  of  reducing  the 
number  facts  to  habit. 

The  third  difficulty  is  adding  a  column  containing  facts  not  in- 
cluded among  the  forty-five  elementary  number  facts.  This  has  been 
fully  discussed  under  series  and  column  addition,  but  under  each  new 
class  of  series  the  teacher  must  emphasize  and  hold  the  child's  atten- 
tion upon  the  new  idea  till  it  is  fully  grasped  and  can  be  applied  with- 
out hesitation.  As  soon  as  this  point  is  reached  and  no  sooner,  the 
child  is  ready  for  another  difficulty.  It  is  the  piling  up  of  difficulty 
after  difficulty  without  mastering  each  in  turn  that  creates  havoc  in 
arithmetic  teaching. 

79 


In  subtraction  the  first  difficulty  is  the  idea  of  subtraction  itself. 
As  already  explained,  this  idea  should  be  brought  out  by  use  of  simple 
problems  within  the  child's  own  experience  and  comprehension.  Do 
not  introduce  the  Austrian  method  till  sure  that  the  subtraction  idea 
is  fully  grasped.  So  too  the  preparation  for  subtraction  must  be  fully 
in  hand  before  formal  subtraction  begins.  Then  these  two  ideas  must 
be  combined;  the  child  finds  the  difference  by  applying  the  prepara- 
tion. If  very  simple  examples  are  first  used,  he  will  see  the  connec- 
tion between  the  two  ideas.  Next  is  the  carrying,  now  used  in  place 
of  the  old  borrowing.  The  method  of  focussing  upon  this  new  diffi- 
culty has  been  fully  discussed. 

As  in  addition  and  subtraction,  so  in  multiplication  and  division, 
focus  the  child's  attention  closely  upon  each  new  idea  in  turn.     The 
steps  have  already  been  indicated,  in  some  cases  merely  by  progres- 
sive type  examples.     These  type  examples  should  be  closely  studied 
by  the  teacher  to  determine  what  is  the  essential  new  difficulty  intro- 
duced in  each.     Particularly  note  the  general  class  of  long  division 
examples  in  which  the  quotient  figure  cannot  be  directly  found  by 
dividing  the  first  digit  or  the  first  two  digits  of  the  dividend  by  the 
first  digit  of  the  divisor.     The  new  difficulty  upon  which  to  focus  here 
is  that  the  subtrahend  cannot  be  larger  than  the  minuend.       To  be 
sure,  this  difficulty  may  have  come  up  incidentally  before,  due  to  a 
child's  carelessness,  but  then  the  teacher  should  merely  have  called 
attention  to  the  particular  error,  and  the  special  difficulty  should  have 
been  left  to  a  later  time,  as  then  some  other  idea  was  under  focus. 
Now  the  teacher  places  on  the  board  several  examples  introducing 
the  new  difficulty  and  questions  the  class  regarding  each  step,   and 
then  when  the  difficulty  is  reached  in  the  example,  she  calls  attention 
to  it  and  asks  the  class  to  watch  hereafter  for  this  trouble.     After 
the  first  example,  the  class  should  see  the  error  as  they  are  looking  for 
just  this.     After  several  examples  have  been  worked  at  the  board  by 
one  child  under  the  direction  of  the  class  and  the  supervision  of  the 
teacher,  the  whole  class  should  be  sent  to  the  board  and  an  example 
dictated  with  special  instructions  to  watch  for     the     new  difficulty. 
Until  the  class  can  work  such  examples  without  effort,   the  teacher 
will  find  it  necessary  to  remind  the  different  individuals  that  they  are 
watching  for  something. 

Similarly  handle  the  principle  that  the  remainder  must  be  small- 
er than  the  divisor. 

In  teaching  reading  and  writing  numbers  above  1000,  focus  on 
the  fact  that  the  digits  between  two  commas  or  to  the  left  of  a  comma 
should  be  read  as  if  they  stood  alone  and  then  the  name  given  as 
thousand  or  million  as  the  case  may  be.  Then  as  a  preparation  for 
the  next  step,  focus  on  the  fact  that  there  must  be  three  figures  be- 
tween any  two  commas  or  to  the  right  of  a  comma.  Emphasize  this, 
calling  attention  to  it  repeatedly. 

80 


Then  in  reading  numbers  in  which  there  are  no  hundreds  or  no 
hundreds  and  tens,  focus  on  the  fact  that  the  digits  or  digit  are  placed 
to  the  right  of  the  period  and  that  the  vacant  places  are  filled  in  with 
ciphers  in  order  to  make  three  places  in  the  period. 

To  teach  writing  such  numbers,  the  teacher  should  write  on  the 
board  25,067  and  have  the  class  at  the  board  copy  this,  then  have 
them  write  dictated  numbers  directly  below  it  in  the  proper  place. 
In  dictating  27,048,  the  teacher  should  say  "27  thousand"  and  pause 
long  enough  for  the  class  to  write  27  and  place  the  comma  after  it. 
Always  hesitate  thus  and  insist  on  the  placing  of  the  comma  at  once. 
As  soon  as  the  class  has  written  the  comma,  continue  the  dictation — 
"48",  letting  the  voice  fall  for  the  first  time  to  indicate  that  the  dic- 
tation is  complete,  and  question  as  to  where  the  digits  must  be  placed. 
Ans.  To  the  right  of  the  period.  Under  what  figures?  Ans.  Under 
67.  How  many  figures  must  there  be  in  a  period?  Ans.  Three. 
Then  what  must  you  do  to  make  three  places?  Ans.  Write  in  a 
naught. 

Similarly  teach  numbers  in  which  two  naughts  must  be  inserted. 
After  a  time  drop  the  questions  but  caution  the  class  to  be  on  the 
watch  for  the  new  point  just  learned.  If  some  member  of  the  class 
still  has  trouble  and  fails  to  insert  the  naught,  question  him  as  be- 
fore. 

In  all  work,  fractions,  denominate  numbers,  decimals,  percent- 
age, proportion,  and  square  root,  follow  the  same  general  plan  of 
focussing  closely  for  some  time  upon  the  specific  difficulty  introduced 
in  each  type  example.     In  the  end  time  will  thus  be  saved. 

In  Proportion  the  order  of  steps  would  be  first  a  preparation 
showing  that  if  the  product  of  two  factors  equals  the  product  of  two 
other  factors,  and  one  of  the  factors  is  missing,  it  may  be  found  by 
dividing  the  product  of  one  pair  of  factors  by  the  factor  in  the  pair 
from  which  the  one  is  missing.  Place  the  two  factors  to  be  multiplied 
above  the  line  and  the  single  factor  below  the  line  and  cancel. 

Introduce  proportion  by  means  of  diagrams  illustrating  the  fact 
that  at  a  given  time  objects  of  different  heights  cast  shadows  of  differ- 
ent lengths;  the  higher  the  object,  the  longer  the  shadow  cast.  Also 
use  any  other  illustrations  that  will  make  the  subject  clear. 

Next  teach  the  statement  of  the  proportion,  emphasizing  that 
like  numbers  should  be  grouped  in  one  ratio.  The  heights  of  the  ob- 
jects should  be  placed  in  one  ratio  and  the  lengths  of  the  shadows  in 
the  other.  Give  much  practice  in  stating  problems  before  attempting 
solution. 

Follow  this  with  the  principle  that  the  product  of  the  means 
equals  the  product  of  the  extremes,  again  giving  practice  in  deter- 
mining what  terms  should  be  multiplied  together.  Then  basing  upon 
the  preparation  mentioned  above,  teach  the  method  of  finding  the 
required  term. 

81 


The  next  difficulty  will  be  inverse  proportion.  Use  a  simple  ex- 
ample as:  If  one  boy  piles  a  certain  amount  of  wood  in  3  hours,  how 
long  will  it  take  two  boys  working  at  the  same  rate  to  pile  the  same 
amount?  Have  the  like  numbers  stated  in  one  ratio,  then  place  the 
odd  term  in  either  the  mean  or  the  extreme  in  order  that  the  result 
may  be  larger  or  smaller  than  the  given  odd  term  as  desired.  That 
is,  if  the  result  should  be  larger  than  the  odd  term,  place  the  odd 
term  so  that  it  will  multiply  the  larger  of  the  other  two  terms;  if  the 
desired  result  should  be  smaller  than  the  odd  term,  place  the  odd  term 
to  multiply  the  smaller  of  the  two  other  terms.  Hereafter  the  first 
thing  to  be  considered  in  a  problem  is,  whether  in  the  new  condition 
the  result  should  be  greater  or  smaller  than  in  the  first  condition 
stated,  then  the  example  should  be  so  stated  that  the  result  will  be 
greater  or  smaller  as  desired. 

If  compound  proportion  is  to  be  taught,  first  put  down  the  ratio 
containing  the  unknown  as  a  mean;  then  state  each  condition  so  that 
the  unknown  will  be  increased  or  decreased  as  the  condition  requires. 
Note  that  in  determining  how  to  state  each  ratio,  its  effect  upon  the 
unknown  alone  is  considered  and  all  other  conditions  are  ignored  for 
the  time  being.  If  the  unknown  will  be  increased  by  a  given  condi- 
tion, place  the  larger  term  in  the  extreme  and  the  smaller  in  the 
mean;  if  the  unknown  will  be  decreased  by  the  condition,  place  the 
smaller  term  in  the  extreme. 

Example:  If  it  takes  50  men  12  days  of  10  hours  each  to  dig  a 
ditch  80  rods  long,  6  feet  wide,  and  4  feet  deep;  how  many  men  work- 
ing 15  days  of  8  hours  each  will  be  required  to  dig  a  similar  ditch  96 
rods  long,  5  feet  wide,  and  6  feet  deep? 

Will  it  take  more  or  less  men  working  15  days  than  it  takes 
when  they  work  12  days?  Ans.  Less.  Then  write  the  smaller  nub- 
ber,  12,  in  th  extreme  to  multiply  the  50,  and  the  15  in  the  mean  to 
divide  the  12  x  50. 

Will  it  take  more  or  less  men  if  they  work  8  hours  a  day  instead 
of  10  hours?  Ans.  More.  Then  state  so  that  the  result  will  be  more; 
that  is,  multiply  the  extreme,  50,  by  the  larger  number  10,  by  placing 
it  in  the  other  extreme.     Where  shall  we  place  the  8? 

Will  it  take  more  or  less  men  to  dig  a  ditch  96  rods  long  than  to 
dig  one  80  rods  long?     Then  place  to  obtain  more. 

Will  a  5  foot  ditch  require  more  or  less  men  than  a  6  foot  ditch? 
Etc. 


50      :      X 


Place  all  the  extremes  above  the  line  and  all  the  means  below 
and  cancel. 


15 

12 

8 

:       10 

80       : 

96 

6 

:        5 

4 

:        6 

82 


The  following  progressive  steps  in  square  root  should  be  empha- 
sized in  turn: 

1.  Mentally  extract  the  square  root  of  such  numbers  as  16;  49, 
81;  100;  144;  400;  etc. 

2.  Mentally  extract  the  square  root  of  the  largest  square  con- 
tained in  any  number  of  one  or  two  digits.  The  square  root  of  the 
largest  square  in  85  is  9;  that  is  the  largest  square  in  85  is  81,  whose 
root  is  9.     The  square  root  of  the  largest  square  in  80  is  8.     Etc. 

3.  Find  the  square  root  by  factoring.  The  square  root  of  36 
equals  the  square  root  of  4x9  =  2x3  =  6.  The  square  root  of  225  = 
the  square  root  of  9x25  =  3x5  =  15.  Or  the  square  root  of  225  = 
the  square  root  of  3x3x5x5  =  3x5  =  15.  Therefore  to  find  the 
square  root  of  a  perfect  square,  find  its  prime  factors  and  take  one 
factor  from  each  pair  of  identical  factors  as  the  factors  of  the  root. 
Find  the  square  root  of  3969.  3969=  (7  X  7)  X  (3  X  3)  X  (3x3), 
therefore  the  square  root  of  3969  is  7  X  3  X  3,  or  63. 

4.  Teach  pointing  off  into  periods  of  two  figures  each  both  ways 
from  the  decimal  point.     567.5780;    5 '83  82. 60  00  00. 

5.  Develop  the  regular  method  of  extracting  the  square  root  of 
integers  of  three  or  four  digits.  Teach  by  means  of  diagram  or  rule. 
See  Stamper,  op.  cit.,  pp.  116-118.  After  developing  method  of  solu- 
tion, use  "chalk  and  talk"  method  with  class  at  board. 

6.  Square  root  of  numbers  of  three  periods. 

7.  Exact  root  of  integer  and  decimal  of  two  and  four  decimal 
places. 

8.  Appropriate  root  found  by  annexing  ciphers  to  the  decimal. 

MONTESSORI  RODS. 

Experiments  in  the  first  grade  along  the  line  suggested  in 
Myers,  Experimental  Psychology,  pp.  72-90,  indicate  that  a  class  will 
learn  a  set  of  facts  logically  grouped  together  as  soon  as,  if  not 
sooner  than,  it  wili  learn  a  single  fact  by  itself.  For  example,  to 
teach  the  facts  whose  sum  is  six,  have  each  child  take  a  six  inch  rod 
and  find  all  the  pairs  of  rods  that  placed  together  will  be  equal  in 
length  to  the  six  inch  rod.  Have  them  write  these  facts  in  the  four 
ways  and  give  various  drills.  The  next  day  have  them  write  or  tell 
all  the  facts  in  the  "six  family"  that  they  can  remember.  Then  have 
them  find  any  forgotten  facts  by  means  of  the  rods.  Before  taking 
up  the  "seven  family",  see  that  they  thoroughly  know  all  previous 
"families".  Children  enjoy  this  work  as  it  appeals  to  the  duzxU 
instinct. 


ERRATA. 


Page  33.  124  8 


6/0)744/0  8  000)65/721—1721 

Page  49,  above  Section  XV,  inverted  figure  7  (£)  should  be  a  cipher. 

Page  41,  fourth  line  from  bottom,     the. 

Page  42,  line  11.     reduced. 

Page  50,  XVIII.     MULTIPLICATION. 

Page  57,  line  4.     5160  should  be  51600. 

XIV.  line  10.     11.52  should  be  115.2. 

Page  61,  See  that  each  .17  in  (b)  is  directly  below  the  corres- 
ponding .17  in  (a). 

Page  71,  line  5  from  bottom,     money. 

Page  77,  line  4  from  bottom,  .005  should  be  .015.  Also  90  should 
be  below  left  hand  column  of  dates  and  35  below 
right  hand  column. 


34 


BIBLIOGRAPHY. 


Bailey.     A  handy  book  on  teaching  arithmetic.     Bailey,  Yonkers,  N. 

Y.     1913. 
Brooks.        The   Philosophy  of  arithmetic.        Normal   Publishing   Co. 

Philadelphia.     1876. 
Brown  and  Coffman.     How  to  teach  arithmetic.      Row  Peterson  and 

Co.     1914. 
Gildemeister.     The  multiplication  tables.     A  Flanagan  Co.     Chicago. 

1905. 
McLellan  and  Dewey.     Psychology  of  number.     Appleton.     1895. 
McMurry.     Special  method  in  arithmetic.     Macmillan.     1905. 
Maxson's  self-keyed  number  cards.     J.  L.  Hammett  Co.,  New  York. 
Shutts.     Handbook  of  Arithmetic.     Ginn. 
Smith.     The  teaching  of  elementary  mathematics.  Macmillan.   1900. 

The  teaching  of  arithmetic.     Teachers  College.     1909. 
Stamper.       A  text  book  on  the  teaching  of  arithmetic.       American 

Book  Co.     1913. 
Stone.      Arithmetical   abilities   and   some   factors   determining  them. 

Teachers  College.     1908. 
Suzzalo.     The  teaching  of  primary  arithmetic.     Houghton,  Mifflin  Co. 

1911. 
Walsh.     Methods  in  arithmetic.     Heath.     1911. 
Woodfield.     A  manual  on  the  teaching  of  division.     Flanagan. 
Young.     The  teaching  of  mathematics.     Longmans.     1906. 

The  teaching  of  mathematics  in  Prussia.     Longmans.      1900. 
Though  many  other  works  have  been  mentioned  in  the  text,  only 
those  dealing  specially  with  the  teaching  of  arithmetic  have  been  in- 
cluded in  the  above  list. 


85 


CONTENTS. 


PART  I. 

General  Principles. 

I.      Reasons  for  the  study  of  method 4 

II.      Aims  of  Arithmetic  teaching 5 

III.  Function  and   extent  of   objective   teaching 5 

IV.  Essentials  for  efficient   recall 5 

V.      Teaching  a  new  number  fact — Drills 6 

VI.      Reviews 7 

VII.     Concert  recitations 8 

VIII.     Attention 8 

IX.      The  art  of  questioning 9 

X.     Class  vs.  individual  recitation 9 

XI.      The  inductive  development  lesson 10 

XII.      Teaching  a  new  operation 11 

XIII.  The  deductive  development  lesson 12 

XIV.  Problem  solving 12 

XV.     Teaching  how  to  study 13 

XVI.     The  study  recitation 14 

XVII.      The  assignment 15 

XVIII.     Dictation 1G 

XIX.     Oral  and  silent  mental  arithmetic 16 


PART  II. 

Primary  Arithmetic. 

1.     First  lessons  in  number 17 

II.     Numbers  above  nine IS 

III.  Reading  and  writing  numbers  above  1000 19 

IV.  Addition 20 

V.      Series  and  column  addition ,  22 

VI.      Subtraction 26 

VII.      Multiplication 28 

VIII.      Division , 30 

IX.      Short  division 31 

X.     Long  division 32 

XI.      Roman  notation 33 

XII.      Cancellation 34 

86 


PART  III. 

Denominate  Numbers  and  Practical  Measurements. 

I.      Denominate  numbers 36 

II.      Square  measure 37 

III.  Cubic  measure 

IV.  Practical  measurements 38 

V.  Perimeter-fences 38 

VI.     Area — acres :;>K 

VII.  Plastering 39 

VIII.  Cement  walks,  paving,  etc 40 

IX.  Painting 40 

X.  Papering 40 

XI.      Carpeting 41 

XII.      Board  measure 44 

XIII.  To  find  the  number  of  bushels  in  a  bin 44 

XIV.  To  find  the  number  of  gallons  in  a  tank  or  cistern.  .  .  41 

PART  IV. 

A.     Common  Fractions. 

I.      Proper  fraction  taught  objectively 45 

II.      Finding  a  fractional  part  of  a  number 45 

III.  The  improper  fraction 45 

IV.  A  fraction  an  indicated  division 45 

V.      To  express  an  integer  as  a  fraction 46 

VI.      Reduction  of  a  fraction  to  an   equivalent  fraction 

with  a  larger  denominator 46 

VII.      Reduction  to  lowest  terms 46 

VIII.      Addition  and  Subtraction  of  simple  fractions  with  a 

common  denominator 46 

IX.      Reduction  of  a  mixed  number  to  an  improper  fraction     47 

X.  Reduction  of  an  improper  fraction  to  a  whole  or  to 

a  mixed  number 47 

XI.      Reduction   to  a     common      denominator  when   one 
denominator  is  a  multiple     of     the  other  and 

addition  and  subtraction  of  such  fractions 47 

XII.  Reduction  to  least  common  denominator  when  two 
or  more  denominators  have  a  common  factor 
and  addition  and  subtraction  of  such  fractions.  .  .      47 

XIII.  Reduction      to      a      common   denominator  when  all 

denominators      are     prime   to  one   another  and 
addition  and  subtraction  of  such  fractions 48 

XIV.  Addition  of  mixed  numbers 48 

A.  Sum  of  two  fractions  equal  to  1. 

B.  Simple  fractions  that  may  be  added  mentally. 

C.  More  difficult  fractions. 

87 


XV. 

XVI. 


XVII. 

XVIII. 

XIX. 

XX. 

XXI. 

XXII. 

XXIII. 

XXIV. 

XXV. 

XXVI. 

XXVII. 


I. 
II. 
III. 

IV. 

V. 

VI. 

VII. 

VIII. 

IX. 


X. 

XI. 

XII. 

XIII. 


XIV. 


Subtraction  of  mixed  numbers 49 

Multiplication  of  a  fraction  by  an  integer 50 

A.  Multiplication  of  numerator. 

B.  Division  of  denominator. 

C.  Cancellation. 

Multiplication  of  an  integer  by  a  fraction 50 

Multiplication  of  a  fraction  by  a  fraction 50 

Multiplication  of  a  mixed  number  by  a  mixed  number  50 

Division  of  a  fraction  by  an  integer 51 

Division  of  an  integer  or  of  a  fraction  by  a  fraction  51 

Division  of  a  mixed  number  by  an  integer 51 

Division  of  an  integer  by  a  mixed  number 51 

Division  of  a  mixed  number  by  a  mixed  number.  ...  52 

Multiplication  of  a  mixed  number  by  a  fraction ....  52 

Multiplication  of  an  integer  by  a  mixed  number.  ...  52 

Division  of  a  mixed  number  by  a  fraction 52 

B.     Decimal  Fractions. 

Meaning 53 

Reading  and  writing 53 

Place  value 53 

Cipher  at  the  right  of  a  decimal 53 

Reduction  of  a  fraction  to  a  decimal 54 

Reduction  of  a  decimal  to  a  fraction 54 

Addition  and  subtraction  of  decimals 54 

Multiplication  of  decimals 54 

Division  of  decimals 54 

A.  Decimal  by  an  integer. 

B.  Decimal  by  a  decimal 55 

Fraction  at  the  end  of  a  decimal 55 

Application  of  decimals  in  division  by  a  mixed  number  55 

Aliquot  parts — Table 56 

Multiplication  by  aliquot  parts 56 

A.  By  aliquot  parts  of  1. 

B.  By  aliquot  parts  of  10,   100,  or  1000 57 

Division  by  aliquot  parts 57 

A.  By  aliquot  parts  of  1. 

B.  By  aliquot  parts  of  10,  100,  or  1000. 


C.     Fractional  Relations. 

I.     To  find  a  fractional  part  of  a  number 57 

II.     To  find  what  part  one  number  is  of  another 57 

III.     To  find  the  whole  when  a  part  and  its  fractional 

relation  the  whole  are  given 58 

D.     Oral  Analysis,  59. 
88 


PART  V. 
Percentage. 

I.      Preparation  for  percentage 60 

II.      Introduction  of  terms  per  cent,  Rate  per  cent,  and  Rate 

III.  Introduction  of  terms  Percentage  and  Base 01 

IV.  Given  the  Base  and  the  Rate  to  find  the  Percentage.  .  61 
V.      Given  the  Base  and  the  Percentage  to  find  the  Rate.  63 

VI.      Given  the  Percentage  and  the  Rate  to  find  the  Base  6 

VII.      Amount 64 

VIII.      Given  the  Amount  and  the  Base  to  find  the  Rate.  .  . 

IX.      Given  the  Amount  and  the  Percentage  to  find  the  Rate  65 

X.      Given  the  Rate  and  the  Base  to  find  the  Amount.  ...  65 

XI.      Given  the  Amount  and  the  Rate  to  find  the  Base ....  6(5 

XII.      Given  the  Amount  and  the  Rate  to  find  the  Percentage  <;7 

XIII.  Given  the  Percentage  and  the  Rate  to  find  the  Amount  67 

XIV.  Difference 67 

XV.     Given  the  Difference  and  the  Base  to  find  the  Rate.  .  .  67 

XVI.      Given  the  Difference  and  the  Percentage  to  find  the  Rate  67 

XVII.     Given  the  Rate  and  the  Base  to  find  the  Difference .  .  68 

XVIII.      Given  the  Difference  and  the  Rate  to  find  the  Base ...  68 
XIX.      Given  the  Difference  and  the  Rate  to  find  the 

Percentage 68 

XX.      Given  the  Percentage  and  the  Rate  to  find  the 

Difference 68 

XXI.      Several  successive  discounts 69 

XXII.      To  find  a  single  Rate  of  discount  equivalent  to 

several  successive  Rates 70 

XXIII.  Marking  goods 70 

XXIV.  Simple  interest 71 

XXV.      Given  the  Interest  and  two  elements  to  find  the 

third  element 72 

XXVI.     Six  per  cent  method 72 

Six  per  cent  blank 73 

XXVII.     Compound  interest 71 

XXVIII.     Bank  Discount 74 

Bank  discount  blank 76 

XXIX.     Discounting  interest  bearing  notes 77 

ADDENDA. 

Prevention  of  copying 78 

Only  one  difficulty  at  a  time 79 

Montessori  rods 83 

ERRATA,  84. 

BIBLIOGRAPHY,  85. 

TABLE  OF  CONTENTS,  86. 

INDEX,  90. 

89 


INDEX 


Accuracy,    importance    of 5 

Acres,    computing 38 

Addition: 

evils   of  counting   in 20 

use  of  Montessori  rods.  .    .  .20,  83 

review   devices 21 

series  and  column  addition 

22-26,   79 

of  fractions 46-48 

of   mixed   numbers 48 

of   decimals 54 

Aims    of    arithmetic    teaching.     .  .    5 

Aliquot  parts 56-57 

Amount 64-67 

Analysis 59 

Area 37-38 

Art  of  questioning 9 

Assignment 15 

Attention 6,   8,   9,   78 

Austrian  method: 

of   subtraction 26 

of   division 31,    54,    84 

Bagley,   quoted: 

on   drudgery 6 

on   statement   of   aim 11 

on   the   assignment 15 

Bank    discount 74-77 

blank 76 

Base 61 

Batavia   system 10 

Bins,    finding  contents  of 44 

Blanks: 

perimeter 38 

acres 39 

plastering 39 

papering 41 

carpeting 43 

6  %   method 73 

bank   discount 76 

Board   measure 44 

Bushels  in  a  bin 44 

Cancellation , 34 

Cards   for  teaching  facts 21 

self -keyed    number 84 

Carpeting 41-43 


Carrying 21,   22,   79 

Cement   walks 40 

"Chalk  and  talk"   method, 

11,   16,  28,  32,  83 

Chalk  ready  at  board 22 

Cipher  at   right  of  decimal 53 

in  writing  numbers 81 

Cisterns,    finding   number   of 

gallons    in 44 

Class  vs.   individual  recitation.    .  .    9 

Column    addition 20,    22-26,    79 

Commission 69 

Concert   recitations 8 

Copying,    prevention    of 78 

Counting 17 

in  addition 20-21 

Cubic    measure 38 

Dates,    subtracting 73 

Decimal    fractions 53-57 

Deductive    development    lesson.     .12 

Definition  vs.   use  of  terms 27 

DeGarmo,   quoted: 

on    art   of    questioning 9 

on   statement   of   aim 10 

Denominate   numbers 36-38 

Development   lessons: 

inductive 10,    41,    65 

deductive 12 

Dictation 16,   80 

Difference 67 

Difficulties,  one  at  a  time 

12,   30,   79-83 

Discount: 

cash 69 

bank 74-77 

Division 30-33 

of  fraction  &  mixed  numbers  51-52 

of  decimals 54 

by   a  mixed   number 55 

by   aliquot   parts 57 

Drills 6,     8 


Eliot,    George,    quoted   on    interest   9 

Erasing  work   discouraged 63 

Errors  corrected  by  child 29 


90 


First    lessons    in    number 17 

Formula: 

for    percentage 62 

for   amount 65 

for    difference 67 

for    marked    price 70 

for    successive    discounts.  ..  .69-70 

for   interest 72 

for   compound    interest 74 

Fractional    relations 57-59 

Fractions 45-52,   55 

Froebel  quoted 5 

Gallons  in   a  tank 44 

Generalization 11,   41,   66 

General    principles 4-16 

Gildemeister  quoted  on  the  order 

of   multiplication    tables..     ..29 

Gordy  quoted  on   reviews 7 

Grube    method,    weakness    of .  .     .  .29 

Hall,    Frank,   quoted  on   careless 

facility 5 

How  to  study,   teaching 12-15 

Indefinite  relative  magnitude..  .17 
Individual  vs.  class  recitation.  ..  9 
Inductive  development  lesson 

10,    41,    65-66 

Interest 6,    9 

Interest,   simple 71 

6'r    method 72 

compound 74 

Knowledge,   kinds  of 7 

Logic 12 

Marking   goods 70 

Maxson's  self-keyed  number  cards  84 

McMurry,    Charles,    quoted: 

on   aim   of  arithmetic 16 

on  the  use  of  books 15 

on   order   of   tables 29 

McMurry,   F.   M.,  quoted: 

on   dependence 13 

on    power   of   initiative 14 

Montessori    rods 20,    83 

Multiplication 28-30 

of  fractions  &  mixed  numbers50-52 


of   decimals 54 

by    aliquot    parts 56 

Munsterberg,   quoted: 

on  old   and  new  facts 7 

on    soft    pedagogy 9 

Number    fact,    teaching    new .  .     .  .    6 
Numbers 17-19 

Objective    teaching 6 

Operations,    teaching    new 11 

Oral    analysis 59 

Oral    arithmetic 16 

Painting 40 

Papering 40 

Paving 40 

Partition 30 

Per  cent 60 

Percentage 60-67 

terms   learned   before   problems   61 
statements  of  relation  and 

solution 58,  59,  63 

Percentage  : 

the    term 61 

several  for  one  base 64 

a   rate  for  each 64 

Perception,    clear,    essential 5 

Perimeter-fences 38 

Place   value 53 

Plastering 39 

Pointing 22 

Practical    measurements 38-44 

Primary    arithmetic 17 

Problems: 

size  of  numbers  in  first 

written 16,   30 

solving 12 

cancellation   in 34,   35 

Proceeds,    net 69 

of  a  note 76 

Proportion 81 

Questioning,    art    of 9 

Quintilian,    quoted 17 

Rate 60 

one    for   each    percentage 61 

Reading  numbers 17-19,   80 

decimals 53 

Reasons    for    study    of    method...    4 


91 


Recall,    essentials  for  efficient.     .  .    5 
Reduction: 

of  denominate   numbers.  .     .  .36-37 

of  fractions 46-48,  54 

of  decimals 54-55 

Reviews ...     .- 7 

Rods  used  in  teaching  addition  20,  83 
Roman  notation 33 


Selling  price:  ^ 

when  there  is  a  gain 64 

when  there  is  a  loss 67 

in    commission    and   discount.     .69 

Series   drill   in   addition 22-26 

Silent   mental   arithmetic 16 

Snow    storm    illustration 7 

Square    measure 37 

Square  root 83 

Study: 

teaching   how   to 13 

recitation 14 


Subtraction 20,    26-28,    80 

of  fractions 46-48 

of   mixed   numbers 49 

of   decimals 54 

Successive    discounts 68-70 

Syllogism 12 

Teachers,  kinds  of 4 

Terms,   use  of  vs.   definition .  .     .  .  27 

Walks,  cement,  on  two  or  more 

sides  of  lot 40 

Waste  in   carpeting 42-43 

Writing: 

numbers 17-20,    80 

decimals 53 

Written  problems,  size  of  numbers 
in   first 16 

Young,  quoted  on  one  step  at  a 

time 12 


M 


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